Preheat oven to 350°F (175°C) and grease a 9x13 inch baking dish.
In a bowl, mix shredded chicken and 1 cup of Monterey Jack cheese.
Divide the chicken mixture evenly among the tortillas, roll them tightly, and place seam-side down in the prepared baking dish.
In a medium saucepan over medium heat, melt the butter. Whisk in flour and cook for 1 minute to form a roux.
Gradually whisk in chicken broth and cook until the sauce thickens and bubbles.
Remove the saucepan from heat and stir in the sour cream, diced green chiles, salt, and pepper until smooth.
Pour the white sauce evenly over the rolled tortillas in the baking dish.
Sprinkle the remaining 1 cup of cheese over the top.
Bake for 20 to 25 minutes, or until the sauce is bubbly and the cheese is lightly browned.