Preheat your oven to 350°F (175°C). Lightly grease a 9x13-inch baking dish.
In a large mixing bowl, combine shredded chicken, 1 cup of shredded Monterey Jack cheese, 1 cup of shredded cheddar cheese, sour cream, and diced green chilies. Stir to combine.
Spoon about 2–3 tablespoons of the chicken mixture into the center of each tortilla. Roll them up tightly and place them seam-side down in the prepared baking dish.
In a medium saucepan, melt the butter over medium heat. Whisk in the flour and cook for about 1 minute, stirring constantly, to create a roux.
Gradually add the chicken broth, whisking constantly to prevent lumps. Cook the sauce until it thickens, about 3-4 minutes.
Remove from heat and stir in 1 cup of sour cream, 1 cup of shredded Monterey Jack cheese, garlic powder, onion powder, salt, and pepper. Mix until smooth.
Pour the white sauce evenly over the assembled enchiladas in the baking dish, making sure they’re fully covered.
Sprinkle the remaining shredded cheese over the top of the enchiladas.
Bake uncovered for 20-25 minutes, or until the cheese is melted and bubbly.
Optional: For a golden, crispy top, broil the enchiladas for 1-2 minutes.
Let the enchiladas cool for 5 minutes before serving. Garnish with fresh cilantro, diced tomatoes, or sliced jalapeños if desired. Serve warm.