Stir together first 6 ingredients in a large bowl. Stir together soups and enchilada sauce in a small bowl.
Spoon one-third of soup mixture into a 13- x 9-inch baking dish coated with cooking spray; top with one-third of tortilla strips. Spoon one-third of soup mixture and half of chicken mixture over tortillas; sprinkle with half of Cheddar cheese. Top with one-third tortilla strips; repeat layers once with remaining soup mixture, chicken, Cheddar cheese, and tortilla strips, ending with tortillas. Sprinkle evenly with Monterey Jack cheese.
Bake at 350° for 40 to 45 minutes or until lasagna is bubbly. Let stand 10 minutes. Serve with desired toppings.
Note: Nutritional analysis does not include toppings.