Preheat oven to 350°F (175°C) and lightly grease a 9x13 inch baking dish.
In a medium bowl, combine the shredded chicken with 1 cup of the Monterey Jack cheese.
In a saucepan over medium heat, melt the butter. Whisk in the flour and cook for 60 seconds to create a light roux.
Slowly whisk in the chicken broth. Continue whisking until the mixture thickens and reaches a simmer.
Remove saucepan from heat. Stir in the sour cream, diced green chiles, cumin, garlic powder, salt, and pepper until smooth.
Place a portion of the chicken and cheese mixture into each tortilla, roll tightly, and place seam-side down in the prepared baking dish.
Pour the creamy queso sauce evenly over the top of the assembled tortillas.
Sprinkle the remaining 1 cup of Monterey Jack cheese over the sauce.
Bake for 20 to 25 minutes until the sauce is bubbling and the cheese is fully melted and lightly browned.