Start by melting the butter in a large saucepan over medium heat. Add flour and whisk constantly for about a minute until it turns into a smooth paste and takes on a light golden hue.

Gradually whisk in the chicken broth, ensuring to stir continuously to dissolve any lumps. Keep cooking for around 3-4 minutes until the sauce thickens up nicely.

Remove the saucepan from heat and stir in the sour cream, cumin, salt, and pepper until completely smooth.

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

Preheat your oven to 350°F while preparing your baking dish. Spread a thin layer of the creamy white sauce in the bottom of a greased 9x13 baking dish.

Take a tortilla, place about 1/3 cup of the chicken mixture down the center, roll it tightly, and place seam-side down in the baking dish. Repeat until all tortillas are filled.

Pour the remaining white sauce evenly over the rolled enchiladas, covering them fully. Top with the rest of the Monterey Jack and cheddar cheese.

Bake uncovered for 25-30 minutes until the sauce is bubbling and the cheese is melted and golden brown. Let it rest for about 5 minutes before serving.