Preheat oven to 350 degrees Fahrenheit (175 degrees Celsius) and grease a 9x13 inch baking dish.
In a medium bowl, combine the shredded chicken and 1 cup of the Monterey Jack cheese.
Distribute the chicken mixture evenly among the flour tortillas, roll them up tightly, and place them seam-side down in the prepared baking dish.
In a medium saucepan over medium heat, melt the butter. Whisk in the flour and cook for 1 minute to form a roux.
Gradually whisk in the chicken broth and simmer until the sauce has thickened and is smooth.
Remove the saucepan from heat and stir in the sour cream, diced green chiles, salt, and pepper until well combined.
Pour the creamy 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.