Preheat oven to 375. Lightly spray a 9×13 baking dish with cooking spray and set aside.
In a large skillet over medium heat, melt butter. Add in the onion and sauté 10 minutes or until softened. Add in the garlic, chili powder, cumin, salt, and pepper. Cook 1 minute.
Sprinkle flour over the onion mixture, stirring constantly, about 1 minute. Whisk in broth and bring to a boil, stirring constantly. Boil for 2 minutes or until thickened then remove from heat.
Add green chilies, diced tomatoes, and sour cream and stir to combine. Add chicken and mix together.
Place a single layer of 1/3 of the tortilla halves (slightly overlapping) to the bottom of prepared baking dish. Top with 1/3 of chicken mixture. Sprinkle with 1/3 of the cheeses. Repeat layers 2 more times.
Cover with oil and bake at 375 for 20 minutes. Uncover and bake 10 more minutes or until lightly browned. Let stand 10 minutes before serving.