Preheat your oven to 350°F (175°C).
In a mixing bowl, combine the shredded chicken, half of the Monterey Jack cheese, half of the cheddar cheese, and the diced green chilies.
In another bowl, blend the sour cream, chicken broth, olive oil, garlic powder, onion powder, cumin, salt, and black pepper until creamy.
Spread a small amount of the creamy sauce across the bottom of a 9x13-inch baking dish.
Spoon approximately 1/4 cup of the chicken mixture onto each tortilla, roll it up tightly, and place seam-side down in the dish.
Pour the remaining creamy sauce over the enchiladas and sprinkle the rest of the cheeses on top.
Bake for 25-30 minutes, or until the cheese is melted and golden brown.
Allow to cool slightly before serving and garnish with fresh cilantro if desired.