Prepare the birria meat if not already cooked. In a large pot, simmer the birria meat in water or broth until tender, which usually takes about 2 hours. Once cooked, shred the meat and set aside.
Heat the vegetable oil in a skillet over medium heat. Sauté the diced onion until translucent, about 5 minutes. Add the shredded birria meat, ground cumin, smoked paprika, salt, and pepper. Mix well and cook for another 5 minutes to combine the flavors.
Preheat your oven to 350°F (175°C).
Warm the corn tortillas in a skillet or microwave to make them more pliable.
Fill each tortilla with a portion of the birria mixture and a sprinkle of cheese. Roll them up and place them seam-side down in a baking dish.
Pour the birria broth over the rolled tortillas, ensuring they are well-coated. Sprinkle the remaining cheese on top.
Cover the baking dish with aluminum foil and bake in the preheated oven for 20 minutes.
Remove the foil and bake for an additional 10 minutes or until the cheese is melted and bubbly.
Once cooked, remove from the oven and let cool for a few minutes. Serve the birria enchiladas hot, garnished with fresh cilantro and lime wedges.