Preheat the oven to 375° F. Spray the insides of a 9x13-inch casserole dish and a 9x9-inch casserole dish with cooking spray. Spread ½ cup salsa verde in the bottom of the larger dish, and spread ¼ cup salsa verde in the bottom of the smaller dish.
For the filling, to a large bowl add ¾ cup salsa verde, chicken, 2 cups shredded jack cheese, 1 cup crumbled queso cheese, sour cream, onion powder, garlic powder, oregano, cumin, salt, chili powder, and black pepper.Stir to combine.
Spread about ½ cup of chicken filling on the bottom half of 1 tortilla, roll it up, and place it into one of the prepared pans on top of the salsa verde. Continue this way until all the filling is rolled (you should get 10 enchiladas). Place 6 enchiladas into the larger casserole dish, and 4 into the smaller dish.
Spread ½cup salsa verde on top of the enchiladas in the larger dish, and spread ¼ cup salsa verde on top of the enchiladas in the smaller dish.
Sprinkle 1 cup shredded jack cheese and ½ cup crumbled queso cheese on top of the enchiladas on the larger dish. Sprinkle ½ cup shredded jack cheese and ¼ cup crumbled queso cheese on top of the enchiladas on the smaller dish.
Cover both dishes with foil. Bake until the cheese is melted and the enchiladas are warm throughout, about 25 minutes. If you want to brown the tops a bit, remove the foil and broil them briefly after baking.