Melt 4 tablespoons of butter in a saucepan over medium heat. Whisk in 1/4 cup of flour and cook for 1 minute until golden. Gradually add 2 cups of chicken broth, whisking until thickened, about 3-5 minutes. Remove from heat and stir in 1 cup of sour cream and 1 teaspoon of cumin. Season with salt and pepper.

In a large mixing bowl, combine 2 cups of shredded chicken, 1/2 cup diced onion, 1 can diced green chiles, and 1/4 cup chopped cilantro. Add 1 cup each of Monterey Jack and cheddar cheese; mix until well combined.

Preheat oven to 350°F (175°C) and grease a 9x13 inch baking dish. Spread a thin layer of white sauce on the bottom. Fill each tortilla with approximately 1/3 cup of chicken mixture and roll tightly. Place seam-side down in the dish.

Pour remaining white sauce over the enchiladas, sprinkle with remaining cheese, and cover with foil. Bake for 25-30 minutes until bubbling and golden, removing foil for the last 5 minutes.

Allow enchiladas to rest for 5 minutes before serving. Garnish with additional chopped cilantro if desired.