In a medium saucepan over medium heat, melt the butter until bubbly. Gradually whisk in the flour, stirring for about 1 minute until it forms a golden roux. Slowly add the chicken broth, whisking continuously for 3–5 minutes until the mixture thickens. Remove from heat, then mix in sour cream, cumin, and season with salt and pepper to taste.

In a large mixing bowl, combine the cooked shredded chicken, diced onion, diced green chiles, and chopped fresh cilantro. Stir in the Monterey Jack cheese and cheddar cheese until everything is evenly incorporated.

Preheat your oven to 350°F and grease a 9x13-inch baking dish. Spread a thin layer of the white sauce on the bottom of the dish. Take a flour tortilla, scoop about 1/3 cup of the chicken filling onto it, roll it tightly, and place it seam-side down in the dish. Repeat this until all tortillas are filled and arranged snugly in the baking dish.

Pour the remaining white sauce generously over the top of the enchiladas and sprinkle the remaining Monterey Jack and cheddar cheese evenly.

Bake uncovered for 25-30 minutes, until the sauce is bubbly and the cheese is golden and melted. Remove from the oven and let rest for about 5 minutes before serving.