In a large saucepan, melt 4 tablespoons of butter over medium heat. Whisk in ¼ cup of all-purpose flour and cook for about 1 minute until a light paste forms. Gradually pour in 2 cups of chicken broth while whisking constantly for 3-4 minutes until the mixture thickens.

Remove from heat and stir in 1 cup of sour cream, 1 teaspoon of ground cumin, salt, and pepper until well combined.

In a mixing bowl, combine the shredded chicken, ½ cup of diced onion, 1 can of diced green chiles, ¼ cup of chopped fresh cilantro, and 1 cup of the Monterey Jack and cheddar cheese mixture; fold gently until evenly distributed.

Preheat your oven to 350°F (175°C). Grease a 9x13 baking dish and spread a thin layer of white sauce on the bottom. Fill each flour tortilla with about ⅓ cup of the chicken filling and roll tightly, placing seam-side down in the baking dish.

Pour remaining white sauce over the enchiladas, sprinkle the remaining 1 cup of cheese on top, and bake for 25-30 minutes until the sauce is bubbling and cheese is golden brown.

Remove the pan from the oven and let the enchiladas rest for about 5 minutes before serving.