Heat oven to 375°F. Lightly oil a 9x13-inch baking dish.
Warm oil in a skillet over medium heat. Soften onion for 3–4 minutes.
Stir in garlic, cumin, and chili powder. Cook until fragrant, about 30 seconds.
Pour in 3/4 cup enchilada sauce and the green chiles. Simmer 1 minute.
Fold in chicken, 1/2 tsp salt, and pepper. Stir until evenly coated and warm.
Microwave tortillas wrapped in a damp towel for 45–60 seconds to soften.
Spread 3/4 cup sauce in the baking dish. Mix half the cheese and half the cilantro into the chicken.
Fill each tortilla with chicken mixture. Roll tightly and place seam-side down.
Top with remaining sauce and cheese. Bake 18–22 minutes, until bubbly and melted.
Rest 5 minutes. Sprinkle remaining cilantro and drizzle lime juice. Serve with sour cream.