Preheat your oven to 350°F (175°C).
In a medium bowl, mix shredded chicken with 1/2 cup of cheese. Divide the mixture evenly between the tortillas, roll them up, and place seam side down in a greased 9x13-inch baking dish.
In a saucepan, melt butter over medium heat. Stir in the flour and cook for 1-2 minutes until bubbling but not browned.
Slowly whisk in chicken broth and cook until thickened, about 3-5 minutes. Remove from heat and stir in sour cream and green chilies. Season with salt and pepper.
Pour the sauce evenly over the enchiladas and sprinkle the remaining cheese on top.
Bake for 20-25 minutes until bubbly, then broil for 2-3 minutes to lightly brown the cheese if desired.
Garnish with fresh cilantro and serve warm.