In a food processor, pulse poblanos, onion, jalapeño, and garlic until they resemble a chunky salsa (10–12 pulses). Scrape down the sides as needed. Transfer mixture to a bowl. Process half the beans and ½ cup broth in the food processor until smooth, about 30 seconds, and set aside.
Pat chicken dry with paper towels and season with salt and pepper. Heat the vegetable oil in a large saucepan over medium-high heat until just smoking. Brown chicken on all sides, about 8 minutes, and transfer to a plate.
Add the chile mixture, cumin, and coriander to the saucepan. Cook over medium heat until vegetables soften, 5–7 minutes. Stir in the pureed bean mixture and 1 1/2 cups of chicken broth. Return chicken and any juices to the pot. Reduce heat to medium-low, cover, and cook, stirring occasionally, for 15–20 minutes until the chicken reaches 160°F. Flip the chicken halfway through.
Transfer the chicken to a cutting board, let it cool slightly, then shred it into bite-size pieces using two forks. Discard the skin and bones.
Stir the remaining whole beans into the chili and simmer uncovered until heated through and slightly thickened—about 5 minutes. Turn off the heat, stir in the shredded chicken, and let sit for 2 minutes. Adjust the consistency with extra broth as needed. Stir in fresh cilantro and lime juice, and season with salt and pepper to taste. Serve hot with your favorite garnishes.