Place 1 can of the drained and rinsed beans in a food processor or blender. Add ¼ cup of the broth and puree until smooth; set aside.
Heat the oil in a large Dutch oven or stockpot over medium-high heat. Add the jalapeño, bell pepper, and onion; sauté until soft, about 7-10 minutes. Add garlic and green chilies; cook, stirring constantly, for 1 more minute.
Stir in the remaining chicken broth, chicken, salsa verde, pureed beans, remaining 2 cans of drained and rinsed beans, cumin, oregano, and thyme. Bring to a boil over high heat.
Once the chili boils, reduce heat to medium-low and simmer, uncovered, stirring occasionally, until the chicken is cooked through and the chili thickens, about 30 minutes. If the chili gets too thick, cover with a lid while it continues to simmer. Taste and season with salt and pepper, if desired.
Add a squeeze of lime juice just before serving and garnish individual bowls with desired toppings.