Process first 4 ingredients in a food processor or blender, stopping to scrape down sides.
Stir together half of Cheddar, half of mozzarella, cream cheese, and next 3 ingredients. Stir in chopped chicken. Spoon mixture evenly onto 1 side of each tortilla; roll up, and place, seam sides down, in a 11- x 7-inch baking dish. Sprinkle with remaining cheeses; top with tomato sauce.
Bake at 350° for 35 minutes. Serve with black beans and rice.
NOTE: If you don't have time to spare for the prep work, you can easily make this dish ahead. Cover and chill enchiladas up to 8 hours before baking, if desired. To serve, uncover and bake at 350° for 40 to 45 minutes or until thoroughly heated.