Combine the chicken, onion, garlic, beans, chicken broth, green chile, cumin, oregano, black pepper, cayenne pepper, and salt in the crockpot. Cover with a lid and cook on LOW heat for 7-8 hours.
When the chicken is tender enough to pull apart, shred it gently, leaving plenty of bite-size pieces. Stir to mix the chicken throughout the soup.
In a separate bowl or large glass measuring cup, whisk together the sour cream and heavy cream until smooth and then pour over the mixture in the crock-pot. Stir to combine and cook on low for another half hour or so, until the soup is warm throughout.
Taste the soup and adjust the salt as desired. Serve with a squeeze of lime and the toppings of your choice.