Season chicken liberally with salt and pepper. Heat oil in large Dutch oven over medium-high heat until just smoking. Add chicken, skin side down, and cook without moving until skin is golden brown, about 4 minutes. Using tongs, turn chicken and lightly brown on other side, about 2 minutes. Transfer chicken to plate; remove and discard skin.
While chicken is browning, remove and discard ribs and seeds from 2 jalapeños; mince flesh.
In food processor, process half of poblano chiles, Anaheim chiles, and onions until consistency of chunky salsa, ten to twelve 1-second pulses, scraping down sides of workbowl halfway through.
Transfer mixture to medium bowl. Repeat with remaining poblano chiles, Anaheim chiles, and onions; combine with first batch (do not wash food processor blade or workbowl).
Pour off all but 1 tablespoon fat from Dutch oven (adding additional vegetable oil if necessary) and reduce heat to medium. Add minced jalapeños, chile-onion mixture, garlic, cumin, coriander, and 1/4 teaspoon salt. Cover and cook, stirring occasionally, until vegetables soften, about 10 minutes. Remove pot from heat.
Transfer 1 cup cooked vegetable mixture to now-empty food processor workbowl. Add 1 cup beans and 1 cup broth and process until smooth, about 20 seconds.
Add vegetable-bean mixture, remaining 2 cups broth, and chicken breasts to Dutch oven and bring to boil over medium-high heat.
Reduce heat to medium-low and simmer, covered, stirring occasionally, until chicken registers 160 degrees (175 degrees if using thighs) on instant-read thermometer, 15 to 20 minutes (40 minutes if using thighs).
Using tongs, transfer chicken to large plate.
Stir in remaining beans and continue to simmer, uncovered, until beans are heated through and chili has thickened slightly, about 10 minutes.
Mince remaining jalapeño, reserving and mincing ribs and seeds (see note above), and set aside.
When cool enough to handle, shred chicken into bite-sized pieces, discarding bones. Stir shredded chicken, lime juice, cilantro, scallions, and remaining minced jalapeño (with seeds if desired) into chili and return to simmer.