Place the chicken in a 6-quart crockpot and add the salt, onion, garlic, cumin, green chiles, beans and broth.
Cover immediately and cook on low for 6 to 7 hours without using a delayed start, until the chicken reaches 165°F.
Transfer the chicken to a clean plate and shred it with 2 forks.
Mash 1 1/2 cups of the beans against the crockpot, then stir in the half-and-half and shredded chicken.
Cover and cook on high for 10 minutes, stir in the lime juice, and serve once the broth is hot but not boiling.