In a medium saucepan over medium heat, melt ¼ cup of butter until it becomes frothy, about 1-2 minutes. Sprinkle in ¼ cup of flour, stirring constantly to create a roux. Cook this mixture for about 1 minute until it starts to bubble gently and turns a light golden color.

Gradually whisk in 3 cups of chicken broth to the roux, ensuring there are no lumps. Continue stirring for 3 to 5 minutes, allowing the mixture to thicken until it coats the back of a spoon. Mix in 4 ounces of diced green chiles and 1 cup of sour cream, then set aside.

In a large mixing bowl, combine 2 cups of cooked, shredded chicken breast, 6 ounces of softened cream cheese, ½ cup of Monterey Jack cheese, and stir well until the mixture is creamy.

Preheat your oven to 350°F (175°C). Spread a thin layer of creamy sauce on the bottom of a 9x13-inch baking dish.

Take one tortilla and scoop about ¼ cup of the chicken filling into the center. Roll the tortilla tightly and place it seam-side down in the prepared baking dish. Repeat until all tortillas are filled.

Pour the remaining creamy sauce evenly over the tortillas and sprinkle the remaining 1½ cups of Monterey Jack cheese on top.

Cover the baking dish with aluminum foil and bake in the preheated oven for 20 minutes. Remove the foil and bake for an additional 5 minutes until the top is golden and bubbling.

Let the casserole cool for about 5-10 minutes before serving to allow the flavors to meld.