Preheat oven to 350°F (175°C) and lightly grease a 9x13 inch baking dish.
In a medium bowl, combine shredded chicken with 1 cup of Monterey Jack cheese, garlic powder, and cumin.
Divide the chicken mixture evenly among the tortillas, roll them tightly, and place them seam-side down in the baking dish.
In a medium saucepan, melt butter over medium heat. Whisk in flour and cook for 1 minute to create a roux.
Gradually whisk in chicken broth and bring to a simmer, stirring constantly until thickened.
Remove sauce from heat and stir in sour cream and diced green chiles until smooth. Season with salt and pepper.
Pour the sauce evenly over the rolled tortillas.
Sprinkle the remaining 1 cup of cheese over the top.
Bake for 20 to 25 minutes until the sauce is bubbling and the cheese is melted and lightly golden.
Allow to rest for 5 minutes before serving.