Preheat oven to 350⁰F. Spray baking dish with nonstick cooking spray.
In a medium sauce pan combine soups, broth, green chilies, onion, canned chili and salsa. Heat thoroughly for about 2-3 minutes.
Add chicken and tortillas. Pour into the prepared pan and sprinkle with cheeses.
Bake covered for 25 minutes, then uncover and bake 45 minutes longer.