Eatchel - The Cub

If you don't have her yet but want to try her out, Eatchel is available as a Trial Operative in Story Mission 12B-9

Eatchel is Snowbreaks resident healer - nobody else even comes close in terms of healing.
With her 5* weapon she can passively grant your other operatives a sizable ATK boost, and her M1 also gives your other operatives a good boost as well, but beyond her logistics Special Talent that's as much as you really need to invest into her - she's not good as a main DPS, having both poor dps and little AoE.
Whilst Eatchel is MVP for casual play due to her healing, at endgame her buffs aren't strong enough to make her an optimal pick for anybody other than Clarina.

Overall

Role: Support - Healer + ATK buffer
Deals Kinetic Damage
(Although ideally doesn't deal damage)
Deals mostly ballistic damage
Mobility is pretty good

Advantages

The strongest healer in the game by a vast margin
Strong passive ATK boost with M1 and her 5* gun
Standard Skill is actually very good for clearing groups of trash mobs

Disadvantages

High-level players usually don't need a healer
Her healing is so absurdly high it trivializes literally all game content, which can make things boring.
Her dmg buffs are outclassed by newer operatives
Low personal DPS and little personal AoE
(Although as a pure-support that hardly matters...)

How to use Eatchel - The Cub

1. Add Eatchel to your team

That's it. Do absolutely nothing else with her - Don't use any of her skills and don't ever swap her in. Just park her on the bench and leave her there.

Standard Skill
Eatchel lunges forward, pulling enemies together in front of her, dealing Kinetic DMG, and recovering some HP.
Her next 2 shots deal AoE damage, have a higher fire rate, and recover more U-Energy.

Whilst her standard skill is good for clearing a group of trash mobs, Eatchel is generally not a good main DPS - if she fails to group up enemies or her 2 special shots fail to kill enemies she just has to run around shooting things with her gun which is completely unbuffed, which is about the least effective way of playing possible.
Her standard skill also doesn't really help her against bosses at all, against which she performs very poorly.

Ultimate Skill
Eatchel's Ultimate Skill deals AoE Kinetic DMG in front of her.
This also restores HP to your entire team.

Using her ult also causes Eatchel to gain Icebound Claws.
Icebound Claws are charged by her ult dealing damage.
Maxing her Icebound Claws does not require much investment.

These Icebound Claws automatically restore HP to operatives who drop below 30% HP.
Icebound Claws also provide the squad with an ATK buff.
The most charged Icebound Claw is automatically lost every 20s.

With M1 and a Kinetic Weapon equipped, Eatchel always has 2 Icebound Claws, and if she has 2 or fewer Icebound Claws they no longer naturally disappear.
This means M1 gives Eatchel an extra passive ATK boost that never disappears.
For a passively healing Eatchel who isn't getting any ATK buffs from other operatives this is generally around a +100-110 ATK buff per Claw.

The number of Icebound Claws Eatchel has can be viewed as the small gauge in the lower left of the screen, to the right of your HP bar and S-Energy gauge.

Support Skill
Whilst off-cooldown: Eatchel passive heals your current operative.
This is a very strong heal - getting over 500HP/s is not overly difficult.

Actually using her Support Skill causes Eatchel to leap towards the target and deal Kinetic DMG, whilst also doing a little healing.
This healing is 12x the healing she does each second whilst this skill is off-cooldown.
This skill also has a 12s cooldown, however it only goes on cooldown after Eatche lreturns from her pounce, which takes 1.1 seconds.
As such, using her support skill does reduced healing compared to her passive healing, although the difference is relatively minor.

Weapon

Blitzing Fangs
Her 5* weapon, and her best weapon.
This gives Eatchel more healing (very good), as well as giving another ATK bonus to the squad, which is excellent.
It also gives Eatchel another passive healing effect, which is great.
If you don't have Blitzing Fangs then any Kinetic Shotgun.
I'll recommend Lapis Lazuli as the best, but don't spend more or go out of your way to get it as it's hardly great - just equip whatever Kinetic Shotgun you have or can get for free.

The game recommends Rock Python as her best 4* weapon, but it isn't - Rock Pythons passive only works if its equipping operative uses a skill, which Eatchel generally doesn't want to do.
You can use Rock Python and actually use Eatchel's Support Skill to get an ATK boost, but that's a false economy - if you don't want Eatchel's passive healing then you shouldn't use Eatchel and should use another stronger support instead.
Even if you did want to use Eatchel's Support skill for healing, she'd be better off using Lapis Lazuli anyway, which is also easier to use.

As such, if you don't have Blitzing Fangs, use whatever Kinetic Shotgun you have lying around - Blitzing Fangs is the only weapon Eatchel can use that actually buffs her passive healing, which is the only reason you're using Eatchel anyway. Blitzing Fangs is also Eatchels best weapon for providing offensive buffs as well.

For weapon Parts, just go with ATK, as her healing scales off her ATK.

Logistics

Use Lux squad - it was designed specifically for her and is her best choice.

For the Special talent, you want +10.6% Healing Bonus (note: Healing Bonus affects healing done, not healing received)
For basic stats, Eatchel's ideal stats are +10.0% ATK and +106 Alignment Index.

If Eatchel does not have her 5* weapon (Blitzing Fangs) then Skill Haste and S-Energy Recovery are also good rolls.
Since you'll ideally get Blitzing Fangs eventually though, it's not worth giving Eatchel non-Alignment rolls, as you'll need to re-roll them into Alignment Index rolls once you eventually get Blitzing Fangs anyway, so you're better off just going for Alignment Index to begin with so you don't waste the re-rolls.

Healing Bonus is the only useful Special stat she can have.
Her healing scales off of her ATK, so ATK is her most useful stat.
Alignment gives her more Healing Bonus, which is good.

Since Eatchel ideally never deals damage, things like Kinetic Damage, Crit rate/dmg, etc are all useless.
U-Energy recovery is not particularly useful for Eatchel to have either.

Eatchels support skill uses 12 S-Energy on a 12s cooldown, so she has no S-Energy issues even if she spams her support skill
If Eatchel has Skill Haste though, she can use more S-Energy than she uses, so without any S-Energy recovery mods she'll eventually run out of S-Energy.
As such, if Eatchel has Skill Haste, a corresponding S-Energy Recovery mod can be useful for longer fights so she doesn't run out of S-Energy.

You can spam Eatchels support skill off-cooldown (if you want). If using Rock Python or Lapis Lazuli this can work well as it procs their buffs, in which case giving Eatchel Skill Haste rolls will allow her to use her support skill more often, increasing the amount of healing she does.
Skill Haste will also increase the uptime on Rock Python's buffs.
With Lapis Lazuli, you only need to use her support skill every 15s to maintain Lapis' buff, but Skill Haste means her support skill comes off cooldown sooner, allowing for more passive healing between support skill casts.
Given that you'll ideally get Blitzing Fangs eventually, it's better to go for Alignment Index to begin with, rather than going for Skill Haste and then needing to re-roll for Alignment Index later when you do get Blitzing Fangs.

If Eatchel has Blitzing Fangs, Eatchel ideally won't use her support skill, and thus is better off with Alignment Index rolls, not Skill Haste or S-Energy rolls.

Healing and Damage Maths: Skill Haste, Alignment Index, and using her Support Skill

Blitzing Fangs:

Eatchel's 5* weapon - Blitzing Fangs, gives an ATK buff based on how much total healing was done in the last 20s, and continuously decreases.
Maintaining the maximum buff from Blitzing Fangs is quite easy with Eatchel's passive healing, and once maxed Blitzing Fang' buff will remain maxed.
If spamming her support skill, then Blitzing Fangs will give a varying ATK buff, as the infrequent healing from Eatchel's support skill means the buff will decrease each second between casts, leading to a fluctuating ATK buff.
This means thta Eatchel's passive healing gives a higher average buff from Blitzing Fangs when compared to spamming her support skill.
This means that Eatchel performs best when she has Blitzing Fangs and is performing passive healing, and never uses her support skill.

You can use her support skill to get a higher Blitzing Fagns buff for the first few seconds of combat, but beyond those first few seconds her passive healing provides the stronger buff.
Lapis Lazuli performs the same regardless of how Eatchel heals, and Rock Python performs best with max Skill Haste, but neither are anywhere near as good as Blitzing Fangs.

If you don't have Blitzing Fangs:

Eatchel's Support Skill passively heals for (5% ATK +37) each second, and using her Support Skill heals for (60% ATK + 446) on a 12s cooldown.
This means that in-theory Eatchel should heal the same regardless of whether she stays off-field to provide passive healing or if she spams her support skill.
In practice though, there are several differences:

Eatchels Support Skill goes off-cooldown only after she returns from her pounce, not when she actually uses her support skill, so using her support skill actually takes over 12s in total, leading to her using her support skill having lower average base healing than her passive healing.
Eatchel needing to return after her pounce adds 1.1 seconds to the total time it takes her to use her support skill.
(This is why the Lux logistics squad removes Winter Fleece stacks 2s after using a skill - it's to allow Eatchel to use her support skill and still have it heal at full WF stacks.)
If you go Skill Haste, then Eatchel can use her Support Skill more often, but if she uses her Support Skill more often than every 10s then she won't be at max Winter Fleece stacks from her Lux logistics squad, leading to reduced healing.
This affects her when she has 2x and 3x Skill Haste rolls, but not 1x Skill Haste.
Similarly, going Skill Haste means she can't take Alignment Index, and thus doesn't gain the extra +6.36% Healing Bonus from each Alignment Index roll.
When given any Skill Haste mods, Eatchel will be using more S-Energy than she recovers, meaning she'll eventually run out of S-Energy.

If spamming her support skill off-cooldown, then giving Eatchel Skill Haste rolls will thus lead to Eatchel doing less-buffed healing more often.

Factoring in the extra time Eatchel takes to return from her pounce, then assuming Lux + Blitzsing Fangs + N6/D4 and doing the maths gives the following average values for how much healing Eatchel does per second, as a percentage of her ATK stat.
The asymptote number on the right shows her average healing after her S-Energy runs out and her healing becomes limited by her S-Energy regen rather than her skill cooldown:
% ATK Logistics rolls Healing method Asymptotes towards

11.57% 3x Alignment Index + 0x Skill Haste rolls Only passive healing (i.e. she never uses her support skill) 11.57% (no change)
10.60% 3x Alignment Index + 0x Skill Haste rolls Spamming her support skill off-cooldown (and thus no passive healing) 10.60% (no change)
12.03% 2x Alignment Index + 1x Skill Haste Spamming her support skill off-cooldown (and thus no passive healing) 11.19%
13.20% 1x Alignment Index + 2x Skill Haste Spamming her support skill off-cooldown (and thus no passive healing) 10.73%
13.84% 0x Alignment Index + 3x Skill Haste Spamming her support skill off-cooldown (and thus no passive healing) 10.03%
11.62% 1x Haste, 1x Alignment, & 1x S-Energy Regen Spamming her support skill off-cooldown (and thus no passive healing) 11.62% (no change)

As such, spamming Eatchels Support Skill can result in her doing more healing on average when compared to her passive healing, but it depends on how she is built.

For longer fights where her healing is limited by her S-Energy Recovery speed, you can either go 3x Alignment Index and let her heal passively, or you can go 1x Haste, Alignment, and S-Energy Recovery and spam her Support Skill off-cooldown.
Both of these perform almost identically in terms of healing.
Whilst the active method has a higher number above, in practice you won't always be able to use her support skill the nanosecond it comes off-cooldown, and the passive healing will deal more on-average because of that.

With 3x Skill Haste rolls, Eatchel uses 12 S-Energy every 8s or so, meaning she will eventually run out of S-Energy, preventing her from spamming her support skill.
Due to the extra 1.1s delay of Eatchel needing to return from her pounce after using her support skill, 1x Skill Haste +1x S-Energy Recovery does allow her to sustain her support skill indefinitely.
You can run 1x Alignment Index, 1x Skill Haste, and 1x S-Energy Recovery mod, which does allow her to indefinitely spam her support skill whilst giving her ok healing.
(Doing the maths, this would give her 11.62%, so this is still technically better than her passive healing, albeit only just.)

When Eatchel runs out of S-Energy, the rate at which her support skill can be used is entirely limited by her S-Energy regen.
If Eatchel has run out of S-Energy, then her best method of healing is just passively healing with 3x Alignment rolls or spamming her support skill with 1 each of Alignment, Skill Haste, and S-Energy Recovery, which perform almost identically.
If you aren't using Blitzing Fangs though then you're better off spamming her support skill to proc her weapon buff (Lapis Lazuli or Rock Python)

Again, whilst using Blitzing Fangs with her passive skill is the best way of using Eatchel, but Skill Haste + Alignment Index + S-Energy are good stats if you don't have Blitzing Fangs yet, athough since you'll want to get 3x Alignment Index when you do get Blitzing Fangs so it's not worth re-rolling for any non-Alignment stats as you'll just need to re-roll them later anyway.

Manifestation

With M1 and a Kinetic Weapon equipped, Eatchel always has 2 Icebound Claws, and if she has 2 or fewer Icebound Claws they no longer naturally disappear.
This means M1 gives Eatchel an extra passive ATK boost for the entire squad.
M2 decreases the cd of Eatchel's ult from 60s to 30s.
Given that Eatchels ult is generally quite weak, you're normally better off using the ult of another operative.
This makes Eatchels ult pretty weak, and hence reducing the cooldown of her ult is not overly useful.
M3 also increases damage by 3% per Claw.
Given that her M1 gives her two claws permanently, this is a 6% damage boost for the squad, which isn't much, although it is entirely passive.
There aren't many squads where you want to use Eatchel's ult, but for those squads M3 is a larger bonus and is a decent buff.
M4 increases the level of Eatchel's ult skill by 1, and increases its healing and damage.
M4 also increases the capacity of Eatchel's Icebound Claws to (35% of her ATK +195), up from (30% +180) (Around 1,680 -> 1,945. Around 15% more.)
15% more Icebound Claw capacity means 15% more ATK buff, so this is a buff to Eatchel's passive ATK buff.
This is not a massive buff to Eatchel's support capabilities, but it does buff them.
M5 increases the max number of Icebound Claws by 1.
This doesn't affect how much she buffs the squad unless she uses her ult, so for most team compositions (where you don't use Eatchel's ult) this isn't useful.

If you want efficiency (good damage with minimal investment), then M1 is a good stopping point.
M3 and M4 do increase Eatchel's capabilities as a support operative, but they are small buffs that require huge investment.
Since Eatchel is generally never an operative whose ult you want to use, M5 is overkill in all but a few niche scenarios or for a few team compositions.

Which Main-DPS to pair with Eatchel?

Eatchel's ATK buffs are much appreciated by almost every every main DPS.
Similarly, since Eatchel doesn't deal any damage, she doesn't get in the way of type-specific buffs like Vidya - Celestial Swan.
This makes Eatchel something of a universal support - she can be used on basically every team.

Eatchel thus works well as a support for all operatives, with only a small number of exceptions:

Clarina - Crimson Moon
Clarina gains extra Skill DMG based upon the amount that she is healed by [from any source], meaning Eatchels healing gives Clarina extra Skill DMG, which means that Eatchel provides Clarina with more buffs than she does any other operative, meaning Eatchel is generally Clarina's best support.
Nita - Pulse
Pulse's damage scales off of her Max HP, not her ATK, which Eatchel doesn't affect.
As such, Eatchel's ATK buffs are useless for Nita - Pulse, although Pulse can still make use of Eatchel's passive healing.
Yao - Nightglow
Nightglow's damage scales off of her DEF, not her ATK, which Eatchel doesn't affect.
As such, Eatchel's ATK buffs are useless for Nightglow, although since Nightglow is only useful as a support operative this isn't much of an issue.
Siris - Freefin
Freefin's buffs scale based off of her Max HP, not her ATK, which Eatchel doesn't affect.
As such, Eatchel's ATK buffs are useless for Freefin, although since Freefin is primarily also a support operative this isn't really much of an issue.