Preheat oven to 350°F (175°C) and grease a 9x13 inch baking dish.
In a medium bowl, combine shredded chicken and 1 cup of Monterey Jack cheese.
Divide chicken mixture evenly among tortillas, roll them tightly, and place seam-side down in the prepared baking dish.
In a saucepan over medium heat, melt butter and whisk in flour to create a roux; cook for 1 minute.
Slowly whisk in chicken broth and continue cooking until the sauce thickens and begins to bubble.
Remove saucepan from heat and stir in sour cream and chopped green chilies; season with salt and pepper.
Pour the creamy sauce evenly over the assembled enchiladas and sprinkle with the remaining 1 cup of cheese.
Bake for 22 to 25 minutes until the cheese is melted and the sauce is bubbling around the edges.