Add the shredded chicken, Monterey Jack, cream cheese, green chiles, salt, and pepper to a mixing bowl. Mix until well combined.
Spoon the chicken mixture evenly into each tortilla and roll them up gently but tightly.
Place each rolled tortilla seam-side down in a greased 9×13-inch baking dish.
In a saucepan over medium heat, melt the butter. Whisk in the flour and cook for 1 minute to form a smooth paste.
Gradually whisk in the chicken broth until smooth. Reduce heat to low and stir in sour cream, green chiles, garlic powder, onion powder, salt, and pepper. Heat gently without boiling.
Pour the white sauce evenly over the enchiladas and sprinkle the remaining shredded cheese on top.
Bake at 350°F (175°C) for 20–25 minutes, until bubbly and fully melted. Let rest for 5 minutes before serving.