Season chicken with salt and pepper. Heat oil in Dutch oven on medium-high. Brown chicken 4 mins per side. Transfer chicken to a plate; discard skin.
Remove ribs/seeds from 1 jalapeño; mince and set aside. Pulse half of poblano, Anaheim chiles, and onions in food processor until chunky; transfer to a bowl. Repeat with remaining chiles and onions; combine.
Pour off all but 1 tbsp of fat from pot; reduce heat to medium. Add minced jalapeños, chile-onion purée, minced garlic, cumin, coriander, and 1/4 tsp salt. Cover and cook, stirring occasionally, until softened, about 10 mins. Remove from heat.
Blend 1 cup cooked vegetables, 1 cup beans, and 1 cup broth in processor until smooth. Return purée to pot with remaining 3 cups broth, remaining can of beans, and chicken. Bring to boil over medium-high; reduce to medium-low, cover, simmer until chicken reaches 160 °F, about 15-20 mins.
Lift chicken onto plate. Stir remaining beans into pot; simmer uncovered until slightly thickened, about 10 mins. Shred chicken, discard bones, return meat to pot.
Stir in lime juice, cilantro, scallions, and reserved jalapeño ribs/seeds for extra heat. Adjust seasoning with salt and pepper; simmer briefly before serving.