Preheat oven to 375°F (190°C).
In a large bowl, combine cooked chicken, half of the Monterey Jack cheese, half of the cheddar cheese, and diced green chilies. Season with salt and pepper.
In a saucepan, melt butter over medium heat. Add onion and garlic and cook until softened.
Stir in flour and cook for 1 minute. Gradually whisk in chicken broth until smooth.
Reduce heat and stir in cream cheese and sour cream until melted and smooth. Season with salt and pepper.
Spread a thin layer of sauce in a 9×13 inch baking dish.
Fill each tortilla with chicken mixture and roll up. Place seam-down in the baking dish.
Pour remaining sauce over enchiladas and top with remaining Monterey Jack and cheddar cheese.
Bake for 25-30 minutes, or until cheese is melted and bubbly.
Let rest for a few minutes before serving. Garnish with fresh cilantro, diced tomatoes, or sliced green onions, if desired.
