In a bowl, mix the chicken, 1 cup of Monterey Jack cheese, garlic powder, salt, and pepper. Set aside.
In a saucepan, melt the butter. Add flour and cook for 1 minute.
Slowly whisk in the chicken broth, sour cream, cream cheese, and green chiles. Simmer until thickened.
Grease a casserole dish. Layer tortillas, then chicken, then sauce. Repeat layers 3 times.
Top with the remaining cheese.
Bake uncovered at 350°F for 30 minutes. Broil briefly to brown the top.