Preheat the oven to 375°F (190°C).
Grease a 9x13-inch baking dish with cooking spray or butter.
Heat olive oil in a skillet over medium heat.
Add chopped onion and cook until soft.
Add minced garlic and cook for 1 minute.
Stir in diced tomatoes with green chilies, chicken, both soups, and sour cream.
Add chili powder, cumin, salt, and pepper. Mix well.
Stir in 1 cup of cheddar cheese until melted and creamy.
Arrange a layer of tortilla pieces in the baking dish.
Spread half of the chicken mixture over the tortillas.
Add another layer of tortillas and the remaining chicken mixture.
Sprinkle with the remaining cheddar and Monterey Jack cheese.
Bake uncovered for 25–30 minutes, until hot and bubbly.
Let the casserole cool for 5 minutes before serving.
Garnish with fresh cilantro or green onions if desired.