Cover chicken with water in a large pot and bring to a boil; reduce heat to low and simmer until chicken is no longer pink in the center, 45 minutes to 1 hour. Remove chicken and set aside to cool. Reserve cooking liquid. Shred chicken when cool enough to handle.

Combine chicken, cooking liquid, white beans, hominy, onion, green chiles, garlic, cumin, oregano, and chicken bouillon in a slow cooker. Cover and refrigerate overnight.

Cook chicken and bean mixture on Low in slow cooker for 4 to 4 1/2 hours. Top individual servings with Monterey Jack cheese.