Preheat your oven to 350°F (175°C) and grease a 9x13-inch baking dish.
In a medium bowl, mix the shredded chicken and 1 cup of Monterey Jack cheese.
Fill each tortilla with the chicken mixture, roll tightly, and place seam-side down in the prepared dish.
In a saucepan, melt butter over medium heat. Whisk in the flour and cook for 1 minute.
Gradually whisk in the chicken broth until smooth and cook until thickened, about 5 minutes.
Remove from heat and stir in the sour cream and diced green chilies. Do not boil.
Pour the sauce evenly over the enchiladas and sprinkle with the remaining cheese.
Bake uncovered for 20-25 minutes, or until bubbly and golden. Serve warm.