Melt butter in a large saucepan over medium heat. Whisk in the flour and cook for 1 minute, stirring constantly, until it forms a smooth paste and turns light golden.

Gradually pour in the chicken broth while whisking constantly to prevent lumps. Continue whisking until the sauce thickens and becomes smooth, about 3 to 4 minutes. Remove from heat and stir in sour cream, cumin, salt, and pepper until fully combined.

In a large mixing bowl, combine shredded chicken, diced onion, green chiles, cilantro, and 1 cup of the combined cheeses. Mix well until evenly distributed.

Preheat oven to 350 degrees F (175 degrees C). Spread a thin layer of white sauce across the bottom of a greased 9x13-inch baking dish.

Place about 1/3 cup of the chicken mixture down the center of each tortilla. Roll tightly and arrange seam-side down in the prepared baking dish.

Pour the remaining white sauce evenly over the rolled enchiladas, making sure all are fully covered. Sprinkle remaining Monterey Jack and cheddar cheese over the top.

Bake uncovered at 350 degrees F for 25 to 30 minutes, until the sauce is bubbling at the edges and the cheese is melted and golden brown.

Remove from oven and let rest for 5 minutes before serving to allow the sauce to set properly.