In a medium saucepan over medium heat, melt 4 tablespoons of unsalted butter until bubbly. Whisk in 1/4 cup of all-purpose flour and cook for about 1 minute until golden. Slowly pour in 2 cups of chicken broth, whisking constantly for 3-4 minutes until the mixture thickens and coats the back of a spoon. Remove from heat, mix in 1 cup of sour cream, 1 teaspoon of cumin, and season with salt and pepper to taste.

In a large bowl, combine 2 cups of cooked shredded chicken, 1/2 cup of diced onions, 1 can of diced green chiles, and 1/4 cup of chopped cilantro. Stir in 1 cup each of shredded Monterey Jack and cheddar cheese until thoroughly mixed.

Preheat your oven to 350°F (175°C) and lightly grease a 9x13-inch baking dish. Spread a thin layer of the prepared white sauce on the bottom of the dish. Warm the flour tortillas in a microwave or on a skillet for a few seconds until pliable, then fill each with 1/4 cup of the chicken mixture, roll tightly, and place seam-side down in the dish.

Once all enchiladas are arranged in the dish, pour the remaining white sauce evenly over the top. Sprinkle the remaining shredded cheese over the sauce. Bake uncovered for 25-30 minutes or until the top is bubbly and golden brown. Allow the enchiladas to cool for 5 minutes before serving.