Melt the butter in a medium saucepan over medium heat. Whisk in the flour, stirring continuously for about 2 minutes until smooth and slightly golden.

Gradually pour in the chicken broth, whisking until the sauce thickens for about 3-5 minutes. Remove from heat and stir in sour cream, cumin, salt, and pepper to taste.

In a large bowl, combine the chicken, onion, green chiles, cilantro, and Monterey Jack cheese. Mix well to create the filling.

Preheat the oven to 350°F (175°C). Spread a thin layer of white sauce in a greased baking dish. Fill tortillas with the chicken mixture, roll them up, and place seam-side down in the dish.

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

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