In a large saucepan over medium heat, melt 3 tablespoons of butter until frothy. Whisk in 3 tablespoons of all-purpose flour to create a roux, cooking for 1 minute until golden and bubbly. Gradually add 2 cups of chicken broth, whisking continuously until the sauce thickens (about 3-4 minutes) to a creamy consistency, coating the back of a spoon.

Remove the thickened sauce from heat and stir in 1 cup of room-temperature sour cream, ½ teaspoon of ground cumin, and salt and pepper to taste. Mix until smooth and creamy.

In a separate bowl, combine 3 cups of shredded chicken, 1 small diced onion, ½ cup of diced green chiles, ¼ cup of chopped fresh cilantro, and 1 cup of the Monterey Jack cheese mixture. Stir well to incorporate all ingredients evenly.

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

Place a medium-sized flour tortilla on a flat surface. Spoon about ⅓ cup of the chicken mixture into the center, roll it up tightly, and place seam-side down in the baking dish. Repeat this process until all the filling is used.

Pour the remaining creamy white sauce generously over the top of the rolled enchiladas. Then, sprinkle with the leftover Monterey Jack and cheddar cheese.

Place the baking dish in the preheated oven and bake uncovered for 25-30 minutes, until hot and bubbly, with a golden top.

Once baked, remove the dish from the oven and allow the enchiladas to cool for 5 minutes before serving.