Preheat oven to 375°F (190°C).
In a saucepan, melt butter over medium heat. Stir in flour and cook for 1 minute.
Slowly whisk in chicken broth. Cook until the sauce thickens—about 3 to 5 minutes.
Remove from heat. Stir in sour cream and green chiles. Add salt and pepper.
In a bowl, mix shredded chicken, onion, and 1 cup of Monterey Jack cheese.
Spoon chicken mixture into tortillas, roll 'em up, and place them seam-side down in a greased baking dish.
Pour the sauce over the enchiladas, sprinkle with the rest of the cheese.
Bake uncovered for 20-25 minutes until bubbly and golden.