Preheat your oven to 350°F (177°C) and spray a 9×13-inch baking dish with nonstick cooking spray.
In a small bowl, mix together the shredded chicken and 1 cup of Monterey Jack cheese.
Fill each tortilla with the chicken and cheese mixture, roll them up tightly, and place seam-side down in the baking dish.
In a skillet over medium heat, melt the butter, whisk in the flour, and cook for about one minute. Gradually whisk in chicken broth and cook until thickened, about 3-5 minutes. Let cool slightly before stirring in sour cream and diced green chilies.
Pour the creamy sauce evenly over the rolled enchiladas and sprinkle with the remaining Monterey Jack cheese.
Bake in the preheated oven for 20-25 minutes until heated through and bubbly. Optionally, broil for an extra minute for a golden top.
Remove from oven, top with sliced green onions, and serve warm.