Preheat oven to 350°F. Brown ground beef and onion in a large skillet over medium heat; drain. Stir in refried beans, enchilada sauce and chilies; set aside.
Lightly grease a 9x13 inch baking dish. Arrange half of the tortilla strips in the bottom of the dish. Top with half of the beef mixture, 1 cup Monterey Jack cheese and 1 cup Cheddar cheese.
Repeat layers and top with remaining cheeses. Bake for 25 minutes or until heated through.
Let stand 10 minutes before cutting into squares to serve.