Preheat your oven to 375°F (190°C) and grease a 9x13-inch baking dish with 1 tablespoon of olive oil.

In a large mixing bowl, combine 3 cups of cooked shredded chicken, 1 teaspoon of garlic powder, 1 teaspoon of onion powder, 1 teaspoon of salt, and ½ teaspoon of black pepper, then add 1 cup of shredded Monterey Jack cheese and stir until blended.

Lay the 8 medium flour tortillas flat and scoop a portion of the chicken mixture onto the center of each tortilla, rolling them up tightly and placing seam-side down in the baking dish.

In a medium saucepan, melt 3 tablespoons of butter over medium heat. Whisk in 3 tablespoons of all-purpose flour for 1 minute, then gradually stir in 2 cups of chicken broth until smooth and slightly thick.

Lower heat and add 1 cup of sour cream and the drained green chilies, stirring until creamy and heated through but not boiling.

Pour the sauce over the assembled enchiladas in the baking dish, ensuring they are completely covered.

Sprinkle the remaining 1 cup of shredded cheddar cheese and leftover Monterey Jack cheese on top.

Bake for 20-25 minutes, or until the cheese is melted and golden-brown. Let it sit for 5 minutes before serving.