Preheat oven to 425 degrees. Spray a 9x13 inch baking pan with cooking spray; set aside. Mix the chicken together with 1 cup of the cheese.
Distribute the chicken and cheese mixture evenly among the tortillas. Roll up and place seam side down in the prepared pan. In a large saucepan, melt the butter. Add the flour; cook and stir for 1 minute.
Add the chicken broth, whisking until smooth. Continue cooking over medium heat until sauce is thick and bubbly. Stir in the sour cream and diced green chiles.
Pour the sauce in an even layer over the enchiladas. Sprinkle with remaining cheese.
Bake for 20 to 25 minutes or until golden and bubbly. Sprinkle with cilantro just before serving.