Melt butter in a large saucepan over medium heat.

Whisk in the flour and cook for about 1 minute until it forms a smooth, golden paste.

Gradually whisk in the chicken broth to the roux, minimizing lumps, until the sauce thickens (about 3-4 minutes).

Remove from heat and stir in the sour cream, cumin, salt, and pepper until well combined.

In a mixing bowl, combine the shredded chicken, diced onion, green chiles, cilantro, and 1 cup of cheese mixture. Mix well.

Preheat your oven to 350°F (175°C).

Spread a thin layer of white sauce on the bottom of a greased 9x13 baking dish.

Place about 1/3 cup of the chicken mixture down the center of each tortilla, roll tightly, and place seam-side down in the dish.

Pour the remaining white sauce over the rolled enchiladas, covering them completely.

Sprinkle with the remaining Monterey Jack and cheddar cheese.

Bake uncovered for 25-30 minutes, or until the sauce bubbles around the edges and the cheese is golden brown.

Let enchiladas sit for about 5 minutes before serving.