In a heavy bottom pot or dutch oven over medium high heat, add olive oil. Add onion and chicken breast meat and allow to brown, flipping the chicken just once. You don't need to cook the chicken through, but you want to get a little bit of a sear on it, about 5 minutes per side. Add minced garlic, give everything a stir, and allow to cook for 2-3 more minutes.
Add 4 cups of the chicken broth (reserving the remaining two cups for later) along with the cumin and green chiles. Allow mixture to come to a boil, and then reduce heat to low. Continue to cook on low uncovered until the liquid reduces by half, about 60-90 minutes. Using two forks, shred the chicken.
Add beans to the pot and stir. To get a thicker chili, smash about half of the beans with your spoon as you stir. Mixture should be a little thick once the beans suck up the remaining liquid. At this point, add the corn and the remaining chicken broth. Stir, add salt to taste, and heat through.