Sauté the shredded chicken, diced onion, diced celery, and butter in a pan until the vegetables are soft.
In a separate bowl, combine the cream of chicken soup, green chilis, and sour cream.
Spoon a little bit of the chili-soup mixture into the bottom of a 9x13-inch baking dish and set the rest aside.
Fill the flour tortillas with the chicken and veggie mixture, roll them up, and place them seam-side down in the baking dish.
Spoon the remaining soup mixture over the tortillas and top with Monterey Jack cheese.
Bake in a 350 degrees F (175 degrees C) oven for about 30 minutes until the top begins to brown.
