Heat the oil in a Dutch oven over medium-high heat, season the chicken with 1 teaspoon salt, and brown it for 4 minutes per side.
Add the onion and cook for 5 minutes, then stir in the garlic and cumin for 30 seconds.
Add the green chiles, beans and broth and simmer for 18 to 22 minutes until the chicken reaches 165°F.
Transfer the chicken to a plate, blend 1 cup of beans with the milk, and stir that mixture back into the pot.
Shred the chicken, return it with the lime juice and remaining salt, and simmer gently for 3 minutes.