In two batches, pulse half of the poblanos, half of the Anaheims, onions, and half of the garlic in a food processor until the consistency of chunky salsa (10–12 pulses). Transfer to a bowl. Process 1 cup of beans and 1 cup of broth in the now-empty processor until smooth (about 30 seconds); set aside.
Pat chicken dry and season with 1/2 teaspoon salt and 1/4 teaspoon pepper. Heat 1 tablespoon oil in a Dutch oven over medium-high heat until just smoking. Brown chicken on all sides (about 8 minutes); transfer to a plate.
Pour off all but 1 tablespoon of fat from the pot (add extra oil if needed). Add the chile mixture, half of the jalapeños, cumin, coriander, and 1/4 teaspoon salt. Cover partially and cook over medium heat, stirring occasionally, until vegetables soften (10 minutes). Stir in the pureed bean mixture and 2 cups broth. Return chicken and juices to the pot and bring to a simmer. Reduce heat to medium-low, cover partially, and cook, stirring occasionally, until chicken registers 160°F (15–20 minutes), flipping halfway.
Transfer chicken to a cutting board, cool slightly, then shred into bite-size pieces with two forks; discard skin and bones.
Stir remaining beans into chili and simmer uncovered until beans heat through and the chili thickens (10 minutes). Stir in shredded chicken, cilantro, scallions, lime juice, and remaining jalapeño. Adjust consistency with extra hot broth as needed. Season with salt and pepper to taste. Serve with desired garnishes.