Preheat oven to 350°F (175°C) and lightly grease a 9x13 inch baking dish.
In a medium bowl, combine the shredded chicken with 1 cup of the Monterey Jack cheese.
Fill each tortilla with approximately 2-3 tablespoons of the chicken mixture, roll tightly, and place seam-side down in the prepared baking dish.
In a saucepan over medium heat, melt the butter. Whisk in the flour and cook for 1 minute to create a roux.
Slowly whisk in the chicken broth and continue cooking until the sauce thickens and reaches a simmer.
Remove the saucepan from the 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 Monterey Jack cheese over the top.
Bake for 20 to 25 minutes, or until the cheese is melted and the sauce is bubbling around the edges.