Place chicken in slow cooker bottom.
Add onion, garlic, chiles, salsa verde, beans, and broth.
Add cumin, oregano, chili powder, cayenne if using, salt, and black pepper; stir gently, keeping chicken submerged.
Cover; cook on low 5-6 hours or high 3-4 hours until chicken is cooked through and tender.
Remove chicken; shred with two forks and return to pot.
Stir in fresh cilantro and lime juice; adjust seasoning to taste.
Serve hot topped with Monterey Jack, avocado, and sour cream, with tortilla chips for dipping.