Heat the oil in a large soup pot or Dutch oven. Add the onion and cook until it starts to soften, about 5 minutes. Stir in the garlic and jalapeño and cook for another minute or two. Stir in the green chiles, cumin, oregano and cayenne pepper. Stir to mix and let it cook for about a minute.
Take the chicken broth and one can of the beans and puree in a blender or food processor until smooth. Add to the pot, along with the remaining 2 cans of beans and the salsa verde. Stir in the chicken. Simmer for 15 minutes, stirring occasionally.
Remove the chili from the heat and stir in the cheese. Serve topped with your desired toppings.
