Prepare the oven and baking dish: Preheat your oven to 350°F (175°C). Lightly grease a 9x13-inch baking dish to prevent sticking.
Mix the filling: In a bowl, combine 3 cups of shredded chicken with 1 cup of shredded cheese. This will create the flavorful filling for the tortillas.
Assemble the enchiladas: Lay a tortilla flat, spoon about 1/3 cup of the chicken mixture onto the center, then roll it up tightly. Place the rolled tortilla seam-side down in the prepared baking dish. Repeat with all the tortillas.
Make the white sauce: In a medium saucepan, melt 3 tablespoons of butter over medium heat. Stir in 3 tablespoons of flour and cook for 1-2 minutes until lightly golden. Gradually whisk in 2 cups of chicken broth, stirring constantly, and cook until the mixture thickens, about 3-4 minutes. Remove from heat and stir in 1 cup of sour cream and 4 ounces of diced green chiles until smooth.
Pour and top: Pour the sour cream white sauce evenly over the assembled enchiladas in the dish. Sprinkle the remaining 1 cup of shredded cheese on top
Bake: Place the dish in the preheated oven and bake uncovered for 20-25 minutes, or until the cheese is melted, bubbly, and lightly golden on top.
Rest and serve: Remove from the oven and let the enchiladas rest for 5 minutes before serving to allow the sauce to set. Enjoy!