Step 1

Preheat oven to 350 degrees F (175 degrees C).

Step 2

Heat oil in a skillet over medium-high heat. Run 1 tortilla through the hot oil until softened, 2 to 3 seconds per side; transfer to a plate. Layer a dry tortilla on top of the cooked tortilla on the plate. Cook another tortilla in the hot oil and stack on the dry tortilla. Repeat process until all the tortillas are used.

Step 3

Cook and stir chicken and onion in the same skillet, adding more oil if needed, until chicken is 2/3 done, about 10 minutes. Add green chiles; cook and stir until chicken is no longer pink in the center and onions are tender, 5 to 10 minutes more.

Step 4

Layer tortillas in a 9x13-inch baking dish; top with chicken mixture, enchilada sauce, Monterey Jack cheese, and mozzarella cheese. Repeat layering until all ingredients are used, ending with sauce and cheese.

Step 5

Bake in the preheated oven until cheese is golden brown, 35 to 45 minutes. Cool until set, 5 to 10 minutes.