In a medium saucepan over medium heat, melt ¼ cup of unsalted butter until it becomes frothy, about 1-2 minutes. Once melted, add in ¼ cup of flour, stirring continuously for about 1 minute to create a smooth roux.

Gradually whisk in 3 cups of chicken broth into the roux, stirring until the mixture thickens, which should take about 3-5 minutes. Once thickened, mix in 4 ounces of diced green chilies and 1 cup of sour cream.

In a large mixing bowl, combine 2 cups of cooked, shredded chicken, 6 ounces of softened cream cheese, 1 teaspoon of cumin, and ½ cup of shredded Monterey Jack cheese. Mix until everything is evenly incorporated and creamy.

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

Take each of the 10, 8-inch tortillas and scoop about ¼ cup of the chicken mixture onto the center. Roll each tortilla tightly and place them seam-side down in the prepared baking dish.

Pour the remaining creamy sauce evenly over the rolled enchiladas in the baking dish. Sprinkle 1½ cups of shredded Monterey Jack cheese on top.

Cover the baking dish with aluminum foil and bake for 20 minutes. Then remove the foil and bake for an additional 5 minutes until the cheese is bubbly and golden brown.