Preheat your oven to 350°F and lightly grease a 9x13-inch baking dish.
In a large bowl, combine the shredded chicken, 1 cup Monterey Jack cheese, 1 cup Pepper Jack cheese, diced green chiles, paprika, salt, and pepper.
Divide the filling among the tortillas. Roll them up tightly and place them seam-side down in the prepared baking dish.
To make the sauce, melt the butter in a saucepan over medium heat. Whisk in the flour and cook for 1 minute.
Slowly whisk in the chicken broth until smooth. Continue cooking until the mixture begins to thicken.
Reduce the heat to low and stir in the sour cream, green chiles, garlic powder, onion powder, and black pepper. Do not boil.
Add the shredded Monterey Jack cheese and stir until melted and smooth. Similar white enchilada sauces use a roux, chicken broth, sour cream, green chiles, and cheese to create a creamy sauce.
Pour the sauce evenly over the enchiladas.
Bake uncovered for 25–30 minutes, or until bubbly and heated through.
Let rest for about 5 minutes before serving.
Garnish with cilantro, diced tomatoes, or jalapeños if desired.