Preheat your oven to 375°F (190°C).
In a mixing bowl, combine the shredded cooked chicken with half of the shredded Monterey Jack cheese and half of the diced green chiles. Stir until well mixed.
In a medium saucepan, melt the butter over medium heat. Whisk in the all-purpose flour and let it cook for about 1-2 minutes. Gradually add in the chicken broth while continuously whisking. Bring to a simmer until it thickens.
Remove the sauce from heat and stir in the sour cream and remaining diced green chiles until smooth.
Lay out the flour tortillas. Spoon the chicken mixture onto each tortilla, roll them up tightly, and place seam-side down in a greased baking dish.
Pour the sour cream sauce over the enchiladas and sprinkle remaining shredded cheese on top.
Bake uncovered for about 20-25 minutes until bubbly and golden brown. Let rest for about five minutes before serving.