Preheat the oven to 350 degrees Fahrenheit (175 degrees Celsius) and lightly grease a 9x13 inch baking dish.
In a medium mixing bowl, combine the shredded chicken and 1 cup of the Monterey Jack cheese.
Place a portion of the chicken mixture into each tortilla, roll tightly, and arrange seam-side down in the prepared baking dish.
In a saucepan over medium heat, melt the butter. Whisk in the flour and cook for approximately 1 minute without browning.
Gradually whisk in the chicken broth and continue cooking, stirring constantly, until the sauce thickens and reaches a simmer.
Remove the sauce from heat and fold in the sour cream and diced green chiles. Season with salt and black pepper.
Pour the creamy sauce evenly over the assembled enchiladas in the baking dish.
Top with the remaining 1 cup of Monterey Jack cheese.
Bake in the preheated oven for 20 to 25 minutes, or until the cheese is melted and the sauce is bubbling around the edges.
Let rest for 5 minutes before serving.