Preheat oven to 350°F (175°C) and lightly grease a 9x13 inch baking dish.
In a large mixing bowl, combine shredded chicken, 1/2 cup of Monterey Jack cheese, sour cream, diced green chiles, cumin, garlic powder, and onion powder.
Warm corn tortillas in a damp paper towel in the microwave for 45 seconds until pliable.
Apportion the chicken mixture equally among tortillas, roll tightly, and place seam-side down in the baking dish.
In a medium saucepan over medium heat, melt butter and whisk in flour to create a blond roux; cook for 60 seconds.
Gradually whisk in milk and chicken broth, stirring constantly until the liquid reaches a simmer and thickens.
Remove the saucepan from heat and whisk in the white American cheese until completely emulsified and smooth.
Pour the queso sauce evenly over the arranged enchiladas and sprinkle with the remaining 1/2 cup of Monterey Jack cheese.
Bake for 20 to 25 minutes until the sauce is bubbling and the internal temperature reaches 165°F (74°C).