First off, there really are only five characters of any import in the story: Cookie, King-Liu…
Who is the Protagonist? I'd have to say Cookie. The story is told through his perspective, he's the one who has the culinary skills which…
First off, there really are only five characters of any import in the story: Cookie, King-Liu, Chief Factor, the Cow, and Thomas. There are additional characters who drop in and out of the narrative, but in terms of who influences the Protagonist's journey, there are only a handful.
Who is the Protagonist? I'd have to say Cookie. The story is told through his perspective, he's the one who has the culinary skills which drive the story's central conceit: Biscuits! He's a sweet soul and rather a simpleton who gets caught up in a scheme conjured up by...
King-Liu. On the face of it, he feels like a Mentor. He certainly talks a good game. He's the one with the business acumen. The one with the vision of what he and Cookie could do with the money, another business venture in San Francisco. He's the one who prods Cookie into their larcenous plan of stealing milk from Chief Factor's cow.
But then, there is a genuine friendship that develops between the pair, which suggests the possibility that King-Liu is an Attractor figure.
Yet it is King-Liu's insistence they go for one last big scam which leads to their demise. A shift from ally to enemy suggests a Trickster.
Given the nuanced approach writer-director Kelly Reichardt brings to her characters, it's probably fair to say that King-Liu is *all* of those archetypes at once: Mentor, Attractor, and Trickster.
An interesting way to look at King-Liu is that he is the physicalization of money. Mentor as entrepreneur. Attractor as wealth. Trickster as greed.
Chief Factor can be seen to be a Trickster: Potential ally in providing Cookie and King-Liu a large sum of money for the tea to be held for an honored guest (the Captain). Enemy when he discovers Cookie and King-Liu have been milking his cow dry.
Cookie's relationship with the Cow would seem to be a Protagonist-Attractor dynamic, except the Cow totally busts Cookie by showing affection to him in the presence of Chief Factor and the Captain. The Cow turns enemy! So maybe a Trickster?
Thomas is barely in the story, but in the critical end part of Act III, it is he who tracks Cookie and King-Liu with his rifle in hand, ready to kill the duo. Even though we don't see him shooting the guys, there is an inference he does. A kind of Nemesis figure perhaps?
Befitting characters in a Kelly Reichardt movie, each is complex. The conventions of mainstream commercial Hollywood storytelling do not pertain with such an indie film. First Cow reminds us that character archetypes are tools, not rules, and that characters are ultimately organic figures, at least in good stories, each with their own multilayered psyches.