Preheat the oven to 350°F and spray a 9x13-inch baking dish with cooking spray.
In a large bowl, combine the shredded chicken, drained spinach, French fried onions, and 1½ cups of the Monterey Jack cheese.
Spoon an equal portion of the chicken mixture onto each tortilla, roll them up, and place seam-side down in the prepared baking dish.
Make the sauce: melt the butter in a small saucepan over medium heat. Whisk in the flour and cook for 1 minute.
Gradually whisk in the chicken broth until smooth, then cook until the sauce thickens.
Remove the sauce from heat and stir in the taco seasoning, sour cream, and diced green chiles until fully combined.
Pour the sauce evenly over the rolled enchiladas and sprinkle with the remaining 1 cup of Monterey Jack cheese.
Bake for 20–25 minutes, then broil for a few minutes if you want the cheese browned. Let rest briefly before serving.
