Preheat oven to 350°F. Spray a 9×13-inch baking dish with nonstick cooking spray.
Make the filling: In a large bowl, mix cream cheese, sour cream, green chiles, garlic powder, onion powder, cumin, salt, and pepper until smooth. Fold in shredded chicken and 1½ cups Monterey Jack cheese.
Make the sauce: Melt butter in a saucepan over medium heat. Whisk in flour and cook 1 minute. Slowly whisk in chicken broth and cook 3–5 minutes, until thickened. Remove from heat and whisk in sour cream, cumin, garlic powder, salt, and green chiles. Return to low heat and stir in 1½ cups Monterey Jack until melted and smooth.
Assemble: Spoon about ⅓ cup filling into each tortilla, roll up, and place seam-side down in the baking dish.
Pour sauce evenly over enchiladas. Sprinkle with cheddar if using.
Bake uncovered for 25–30 minutes, until bubbly. Rest 10 minutes before serving. Garnish if desired.