Preheat oven to 425. Spray a 9x13 glass baking dish with cooking spray and set aside.
In medium-sized mixing bowl, combine shredded chicken, 1 cup of cheese, and onion. Divide the mixture evenly between the tortillas, placing the meat mixture down the middle of each tortilla. Roll each tortilla and place seam-side down in baking dish.
3 cups chicken, 2 cups Jack cheese, ½ onion, 10 flour tortillas
In a medium-skillet melt butter then stir in the flour and cook for one minute. Add chicken broth, and salt and pepper, whisking until smooth. Cook until sauce thickens and bubbles, stirring frequently. Add in sour cream and green chilies.
3 Tablespoons butter, 3 Tablespoons all-purpose flour, 2 cups chicken broth, 1 cup sour cream, 7 ounces diced green chilies
Pour sauce over the top of the enchiladas. Top with remaining cheese and bake for 425 for 20-25 minutes.
green onions