Preheat the oven to 375°F (190°C).
In a medium skillet, heat the vegetable oil over medium heat. Add the chopped onion and cook until softened, about 5 minutes.
Add the minced garlic, ground cumin, chili powder, smoked paprika, cayenne pepper, salt, and black pepper to the skillet. Cook for 1 minute until fragrant.
Stir in the diced green chilies and 1/2 cup of the enchilada sauce. Cook for 2 minutes, then remove from heat.
In a large bowl, combine the shredded chicken with the cooked onion and chili mixture and 1 cup of shredded cheddar cheese. Mix well.
Warm the corn tortillas in a dry skillet or microwave for about 20 seconds to make them pliable.
Spread 1/2 cup of the enchilada sauce evenly over the bottom of a 9×13 inch baking dish.
Place about 1/3 cup of the chicken mixture onto each tortilla, roll up tightly, and place seam side down in the baking dish.
Pour the remaining enchilada sauce evenly over the rolled tortillas.
Sprinkle the remaining 1 cup cheddar cheese and 1 cup Monterey Jack cheese evenly on top.
Bake in the preheated oven for 20-25 minutes, or until the cheese is melted and bubbly.
Remove from the oven and let cool for 5 minutes.
Garnish with chopped fresh cilantro and serve with sour cream and lime wedges.