Mix shredded chicken, sour cream, cheeses, garlic powder, onion powder, salt, pepper, and cilantro in a bowl.
Spoon filling into tortillas, roll tightly, and place seam-side down in a baking dish.
Melt butter in a saucepan and whisk in flour to create a roux.
Slowly whisk in chicken broth and cook until slightly thickened.
Stir in sour cream, cheese, salt, and pepper to form a creamy white sauce.
Pour sauce over enchiladas, top with cheese, and bake at 375°F for 20–25 minutes.
Garnish with cilantro and green onions before serving.