Preheat the Oven: Preheat the oven to 350°F (175°C) while preparing the enchiladas.
Prepare the Chicken Filling: In a mixing bowl, combine shredded chicken with 1 cup of shredded Monterey Jack cheese.
Fill and Roll the Tortillas: Spoon the chicken and cheese mixture into each tortilla, roll tightly, and place seam-side down in a greased 9x13 inch baking dish.
Make the Roux: In a medium saucepan, melt butter over medium heat, add flour, and stir continuously to form a smooth paste; cook for 1-2 minutes.
Add Chicken Broth: Gradually pour in chicken broth, stirring constantly until the mixture thickens and bubbles.
Incorporate Sour Cream and Green Chilies: Remove from heat, stir in sour cream and diced green chilies until well combined.
Pour Sauce Over Enchiladas: Pour the creamy sauce evenly over the enchiladas in the baking dish.
Top with Remaining Cheese: Sprinkle the remaining 2 cups of shredded Monterey Jack cheese over the enchiladas.
Bake in the Oven: Bake for 20-23 minutes until heated through and the cheese is melted and bubbly.
Broil for Crispy Top (Optional): Broil for an additional 2-3 minutes for a golden, crispy topping if desired.
Serve Warm: Let cool slightly before serving; garnish with cilantro, diced tomatoes, or avocado slices if desired.