This is a wonderful, healthy chicken chili. Don't be scared by how many green chilies are in this recipe. I don't like a lot of heat and this is what I term mild. The raw jalapenos added at the end contain the heat, so beware!
Remove and discard ribs and seeds from 2 jalapeños; mince. In food processor, process half of the poblano chiles, Anaheim chiles, and onions until consistency of chunky salsa, ten to twelve 1-second pulses. Transfer mixture to another bowl. Repeat with remaining poblano chiles, Anaheim chiles, and onions; combine with first batch (do not wash food processor blade or workbowl).
Add 1 tablespoon vegetable oil to Dutch oven with heat on 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 bowl. 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 to Dutch oven and bring to boil over medium-high heat. Reduce heat to medium-low and simmer, covered, stirring occasionally, for about 10 minutes.
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, and set aside. Stir lime juice, cilantro, and remaining minced jalapeño (with seeds if more heat is desired) into chili and return to simmer. Adjust seasonings with salt and pepper and serve. Top with monterey jack cheese, if desired.