Place chicken in the bottom of a 6-quart or larger slow cooker. Top with the chicken broth, white beans, green chiles, garlic, onion, cumin, oregano, salt, and cayenne. Stir to combine. Cover and cook on low for 4-6 hours or high for 2-4 hours, until the chicken is cooked through.

Remove the chicken breasts to a plate. Once cool enough to handle, shred and set aside.

With an immersion blender, puree a portion of the chili to thicken it, leaving some of the beans whole. (You can also transfer a few ladlefuls of the chili to a food processor and roughly blend, then stir the blended portion back into the chili.)

Stir in the shredded chicken and cilantro. Portion into bowls and top with a squeeze of fresh lime juice. Add any other desired toppings and enjoy.