Cook the chicken using your preferred method with onion, garlic, 1/2 teaspoon cumin, and 1/2 teaspoon Mexican oregano until fully cooked and tender.
Shred, dice, or slice the cooked chicken and place it in a bowl.
Season the chicken with the remaining 1/2 teaspoon cumin, 1/2 teaspoon Mexican oregano, and 1/2 teaspoon ground pepper, then mix well.
In a large bowl, combine cream of mushroom soup, cream of chicken soup, diced green chilies, sour cream, milk, cilantro, remaining seasonings, and 1 1/2 cups of the Mexican mixed cheese until smooth.
Preheat the oven to 375°F (190°C).
Spread a thin layer of the soup mixture in the bottom of a 13×9-inch baking dish.
Spread about 1 tablespoon of sauce onto a tortilla, add seasoned chicken, roll tightly, and place seam-side down in the baking dish.
Repeat with remaining tortillas until the dish is full.
Pour the remaining sauce evenly over the enchiladas.
Sprinkle Monterey Jack pepper cheese over the top.
Bake for 30 minutes until bubbly and lightly golden.