Preheat the Oven:
Preheat your oven to 350°F (177°C) and spray a 9×13-inch baking dish with nonstick cooking spray. Set aside.

Prepare the Filling:
In a small bowl, mix together the shredded chicken and 1 cup of Monterey Jack cheese.

Assemble the Enchiladas:
Fill each tortilla with the chicken and cheese mixture, then roll each one up and place it seam-side down in the prepared baking dish.

Make the Sauce:
In a skillet, melt the butter over medium heat. Sprinkle the flour over the melted butter and whisk to combine. Cook for about 1 minute to remove the raw flour taste.
Remove the skillet from heat and slowly whisk in the chicken broth. Return to the heat and cook until the mixture has thickened and is bubbly, about 3-5 minutes.
Let the sauce cool for a few minutes before adding the sour cream to prevent curdling. Stir in the sour cream and diced green chilies until the sauce is smooth.

Pour the Sauce:
Pour the sauce evenly over the enchiladas in the baking dish. Sprinkle the remaining 2 cups of Monterey Jack cheese on top.

Bake the Enchiladas:
Bake in the preheated oven for 20-25 minutes, or until the enchiladas are heated through and the sauce is bubbly. If you want a golden top, turn on the broiler for 1-2 minutes until the cheese is lightly browned.

Garnish and Serve:
Once baked, top the enchiladas with sliced green onions and serve warm.