Eatchel - The Cub

If you don't have her yet but want to try her out, she 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 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
Deals mostly ballistic damage
Mobility is about standard

Advantages

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

Disadvantages

High-level players usually don't need a healer
Her dmg buffs are outclassed by newer operatives
Low personal DPS and little personal AoE
(Although as a pure-support it 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.

Standard Skill
Eatchel lunges forward, dealing Kinetic DMG and recoering 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, 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.
For a passively healing Eatchel who isn't getting any ATK buffs from other operatives this is generally around +100-110 ATK per Claw.

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 Ability causes Eatchel to leap towards the target and deal Kinetic DMG, whilst also doing a little healing.
This healing tends to be around 6,000 with max investment.

You don't want to ever use Eatchel's Support Ability - just don't use it so it stays off-cooldown and thus Eatchel gives her passive healing, which is much more useful.

Weapon

Blitzing Fangs
Her 5* weapon, and her best weapon.
This gvies 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 - 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 are its equipping operative uses a skill, which Eatchel should never 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 [were so insane that you] did want to use Eatchels Support skill for healing, she'd be better off using Lapis Lazuli anyway.

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

For weapon Parts, just go with ATK.

Logistics

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

For stats you want +10.0% ATK and +106 Alignment Index.
For the Special talent, you want 10.6% Healing Bonus

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

Since Eatchel never deals damage, things like Kinetic Damage, Crit rate/dmg, etc are all useless.
Since she never uses a skill, Skill Haste is also useless, as are S-Energy and U-Energy regen.

ATK, Alignment Index, and Heal Bonus are the only three stats Eatchel can actually use.

If using Lapis Lazuli:
If Eatchel doesn't have Blitzing Fangs yet and is using Lapis Lazuli, then Skill Haste can be a cope stat. It's bad, but it does actually do something.
Eatchels Support Skill has a 12s cd, whereas Lapis' buff lasts 15s, so whilst the buff has 100% uptime regardless, having Eatchel's Support Skill come off-cooldown quicker means she heals more, so Skill Haste would translate to more healing if and only if using Lapiz Lazuli and re-applying Lapis Lazuli's buff every 15s.
You'd need to re-roll the Skill Haste when you got Blitzing Fangs though, so best to avoid Skill Haste altogether so you don't need to re-roll.

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.
M2 decreases the cd of Eatchel's ult from 60s to 30s.
M3 also increases damage by 3% per Claw.
This is a 6% damage boost for the squad, which isn't much, although it is entirely passive.
M4 increaes 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.
M5 increases the max number of Icebound Claws by 1.

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 Eathcel is generally never an operative whose ult you want to use, M5 is overkill in all but a few niche scenarios.

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 works well as a support for all operatives, with only three real exceptions:

Clarina - Crimson Moon
Clarina gains extra Skill DMG based upon the amount that she is healed by [from any source], meaning Eatchel 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 damge scales off of her DEF, not her ATK, which Eatchel doesn't affect.
As such, Eatchel's ATK buffs are useless for Yao - Nightglow, although since Nightglow is also a support operative this isn't really much of an issue.