Preheat oven to 375°F (190°C). Lightly oil a 9x13 baking dish.
Sauté onion in olive oil until softened, then stir in shredded chicken, cumin, half the salsa verde, salt, and pepper; heat through.
Spoon chicken mixture down the center of each tortilla, add a little cheese, roll snugly, and place seam-side down in the dish.
Mix remaining salsa verde with sour cream and pour evenly over the enchiladas; sprinkle with remaining cheese.
Bake 18–22 minutes until bubbly and golden at the edges. Garnish with cilantro and lime before serving.