In a large saucepan over medium heat, melt the butter until frothy. Whisk in the all-purpose flour and cook for about 2 minutes until lightly golden. Gradually pour in chicken broth, whisking continuously until thickened and smooth, about 5-7 minutes. Remove from heat and blend in sour cream, ground cumin, and salt and pepper to taste.

In a mixing bowl, combine cooked shredded chicken, half a diced onion, diced green chiles, and Monterey Jack cheese. Stir until evenly coated.

Preheat the oven to 350°F (175°C) and grease a baking dish. Spread a thin layer of white sauce on the bottom of the dish. Fill a flour tortilla with the chicken mixture, roll tightly, and place seam-side down in the dish. Repeat until all tortillas are filled.

Pour the remaining white sauce over the enchiladas and sprinkle with remaining Monterey Jack and cheddar cheese. Bake uncovered for 25-30 minutes, until bubbly and golden brown.

Let the enchiladas rest for about 5 minutes before serving.