Heat the olive oil in a large pot over medium heat. Add the onion and cook, stirring, until slightly soft, about 3 minutes. Add the garlic, chopped jalapeno, green chiles and cumin and cook, stirring, until the cumin is toasted, about 2 minutes. Stir in the chicken broth, chicken and beans along with the liquid from one of the cans of beans. Bring to a boil, then reduce the heat to medium low and simmer, stirring occasionally, until thickened, about 15 minutes.
Using a potato masher, mash the chili until about half of the beans are broken up. Stir in the corn, cilantro and 1/2 cup cheese. Divide the chili among bowls and top with the remaining 1/2 cup cheese and the sliced jalapeno. Serve with sour cream and cornbread, if desired.