In a medium saucepan over medium heat, melt the butter. Whisk in the flour and cook until golden, then gradually add chicken broth while whisking until thickened. Stir in sour cream, cumin, salt, and pepper.

In a large bowl, combine shredded chicken, onion, green chiles, cilantro, and Monterey Jack cheese. Mix until creamy.

Preheat oven to 350°F (175°C). Spread a thin layer of white sauce in a greased baking dish. Roll tortillas with filling and place seam-side down. Repeat until all are arranged in the dish.

Pour remaining white sauce over enchiladas and sprinkle with remaining cheddar cheese. Bake for 25-30 minutes until bubbly and golden.

Let enchiladas rest for 5 minutes before serving. Garnish with additional cilantro if desired.