Heat 1 tablespoon butter in a large pot or Dutch oven over medium heat. Pat 2 chicken breasts dry then season all over with 2 teaspoons kosher salt.
Place chicken in pot, skin-side-down and cook, without moving, until deeply browned, 5–6 minutes. Using tongs, transfer chicken to a plate.
Add 2 diced poblanos and 1 diced onion to drippings, season with ½ teaspoon salt and cook, stirring frequently, until softened and starting to brown, about 5 minutes.
Stir in 2 tablespoons garlic, 1 tablespoon coriander, and 1 tablespoon cumin; cook 1 minute. Stir in 2 cans green chiles and 5 cups water. Add browned chicken, and any juices, back to pot; bring to a boil then reduce to a simmer and cook, partially covered, until chicken is coked through (165ºF) 15–18 minutes.
Using tongs, remove chicken breasts and transfer to a plate or sheet pan; let cool.
Meanwhile, combine 1 can of drained beans, 1 cup cilantro and 1 cup broth/vegetable soup mixture (from pot) in a blender; purée until completely smooth.
Once chicken is cool enough to handle, remove and discard skin. Remove meat from bones and shred meat into bite-sized pieces using two forks. Discard bones.
Add chicken, remaining can of beans and puréed mixture back to pot. Bring back to a low simmer; stir in 2 tablespoons lime juice and season to taste with additional salt.