Preheat the oven to 350°F (175°C).
In a medium bowl, combine the shredded chicken, half of the salsa verde, chopped onion, cilantro, olive oil, ground cumin, garlic powder, salt, and black pepper. Mix well to combine.
Warm the corn tortillas in a dry skillet over medium heat for about 20 seconds on each side or until pliable, or wrap them in a damp paper towel and microwave for 30 seconds.
Spread 1/2 cup of salsa verde evenly on the bottom of a 9×13 inch baking dish.
Place about 1/3 cup of the chicken mixture in the center of each tortilla, sprinkle with a little of the Monterey Jack cheese, then roll up the tortilla tightly and place seam side down in the baking dish.
Repeat with the remaining tortillas and chicken mixture until the baking dish is full.
Pour the remaining salsa verde evenly over the top of the enchiladas.
Sprinkle the shredded Monterey Jack and sharp cheddar cheese evenly over the salsa verde.
Cover the baking dish tightly with aluminum foil and refrigerate if making ahead. When ready to bake, preheat oven to 350°F (175°C) and bake covered for 20 minutes.
Remove the foil and bake for an additional 10 minutes or until the cheese is melted and bubbly.
Let the enchiladas rest for 5 minutes before serving.
Serve with lime wedges and sour cream if desired.