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

Mix cornmeal, cumin, chile, and salt together in a bowl. Beat egg in a shallow bowl. Dip each piece of chicken in the cornmeal, then in the egg, then back in the cornmeal.

Heat oil in a large pan over medium heat. Cook chicken until browned, about 5 minutes per side. Transfer chicken to a baking dish and top with sliced Monterey Jack cheese. Cover with chile sauce.

Bake in the preheated oven until chicken is no longer pink in the center, about 20 minutes. Serve each piece with some crumbled cotija on top.