Preheat the oven: Heat oven to 375°F (190°C). Lightly grease a 9x13-inch baking dish.
Make the filling: In a large bowl, combine shredded chicken, 1½ cups Monterey Jack cheese, green chiles, sour cream, diced onion, garlic powder, cumin, salt, and black pepper. Stir until well combined.
Make the sauce: Melt butter in a saucepan over medium heat. Add garlic and cook for about 30 seconds.
Whisk in the flour and cook for 1 minute. Gradually whisk in the chicken broth until smooth. Cook for 3–5 minutes, stirring occasionally, until slightly thickened.
Finish the sauce: Reduce heat to low. Stir in sour cream, 1 cup Monterey Jack cheese, cumin, salt, and pepper. Stir until smooth and the cheese has melted. Do not boil the sauce after adding the sour cream.
Prepare the dish: Spread about ½ cup of the sauce over the bottom of the prepared baking dish.
Fill the tortillas: Warm the tortillas briefly so they are flexible. Add about ⅓ cup of chicken filling to the center of each tortilla. Roll tightly and place seam-side down in the baking dish.
Add sauce and cheese: Pour the remaining white sauce over the enchiladas. Sprinkle the remaining 1 cup of shredded cheese evenly over the top.
Bake: Bake uncovered for 20–25 minutes, until hot, bubbly, and the cheese is melted.
Serve: Let the enchiladas rest for 5–10 minutes. Garnish with cilantro and green onions, if desired.