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1. Cover beans with 1 gallon (4 quarts) water. Add 1/4 cup salt and stir until dissolved. Cover and let rest at room temperature at least 8 hours and up to 24. Drain and rinse beans.
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2. Adjust broiler rack to 8 inches below broiler element and preheat broiler to high. Place poblanos, Anaheims, jalapenos, onion, and garlic on a foil-lined rimmed baking sheet. Toss with one tablespoon oil using your hands to coat. Season with salt. Place under broiler and broil, turning peppers and rearranging vegetables occasionally, until peppers are blackened on all sides and skins are wrinkled all over, 15 to 20 minutes total. Gather up foil and form a sealed pouch. Let chilies rest for 5 minutes.
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3. Place chilies and chicken stock in a large bowl. Peel chilies under the chicken stock, leaving skins and seeds behind. Transfer chili flesh to the cup of a hand blender or a standing blender. Add broiled onion, broiled garlic, and the canned jalapeno (do not add jalapeno pickling liquid). Blend until a smooth puree is formed. Set aside.
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4. Heat remaining oil in a large Dutch oven over medium heat until shimmering. Add cumin and coriander stirring, until fragrant, about 30 sec.. Add chili puree and cook, stirring, until incorporated. Strain chicken stock into pot, pressing on skins and seeds to extract liquid. Discard skins and seeds.
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5. Add soaked beans and chicken breasts to pot, adding water until beans and chicken are fully submerged. Bring to a boil, reduce to a bare simmer and cook, stirring occasionally, until chicken breasts register 150u00b0F on thermometer, about 15 min.
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6. Transfer chicken breasts to a bowl and let rest. Continue simmering broth and beans until fully tender, about 1 hr. Remove 1 1/2 cups of beans and liquid and transfer to a standing blender. Blend until smooth. Stir back into pot.
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7. Shred chicken into bite-sized pieces and stir into stew. Stir in half of cheese until melted. Stir in lime juice and 1/2 cilantro.
