Preheat oven to 350°F (175°C).
In a large bowl, combine shredded chicken, 1 cup salsa verde, chopped onion, cilantro, sour cream, cumin, garlic powder, salt, and black pepper. Mix well.
Heat vegetable oil in a skillet over medium heat. Lightly fry each tortilla for about 10 seconds on each side until pliable. Drain on paper towels.
Spread 1/2 cup salsa verde evenly on the bottom of a 9×13 inch baking dish.
Spoon about 1/4 cup of the chicken mixture onto the center of each tortilla, then sprinkle with a little Monterey Jack cheese.
Roll up each tortilla tightly and place seam side down in the baking dish.
Pour the remaining salsa verde evenly over the rolled enchiladas.
Sprinkle the remaining Monterey Jack cheese and all the cheddar cheese evenly over the top.
Cover the dish tightly with plastic wrap or aluminum foil and refrigerate if making ahead, up to 24 hours.
When ready to bake, preheat oven to 350°F (175°C) if not already heated.
Remove plastic wrap or foil and bake uncovered for 25 to 30 minutes, or until cheese is melted and bubbly.
Let cool for 5 minutes before serving. Serve with lime wedges.