In a mixing bowl, combine the shredded rotisserie chicken with half of the taco seasoning, softened cream cheese, shredded Monterey Jack cheese, and diced chilies. Mix well and set aside.

Preheat your oven to 350°F (175°C). Spoon the chicken mixture onto each warmed flour tortilla, roll them up tightly, and place seam-side down in a greased baking dish.

In a skillet, melt butter over medium heat. Add the remaining taco seasoning and flour, stirring for about 3 minutes. Gradually whisk in chicken broth until smooth and thicken. Remove from heat and fold in the remaining shredded cheese and sour cream.

Pour the sauce over the enchiladas, ensuring they are covered. Bake for 25 minutes until golden brown.

Allow to cool for a few minutes before serving. Garnish with fresh cilantro if desired.