Start by preparing the birria. In a large pot, combine the beef chuck roast, chopped onion, minced garlic, and beef broth.
In a saucepan, bring water to a boil and add the dried guajillo and ancho chiles. Boil for about 10 minutes until softened.
Drain the chiles and add them to a blender along with cumin, oregano, salt, and pepper. Blend until smooth, adding a little beef broth if needed.
Pour the blended sauce over the meat in the pot and mix well. Cover and simmer on low heat for about 1.5 hours until the meat is tender.
Once the birria is ready, shred the meat with two forks and set aside. Reserve about 1 cup of the broth.
Heat a skillet over medium-high heat and lightly fry the tortillas, one at a time, for about 30 seconds on each side until they are pliable.
Preheat your oven to 350°F (175°C).
To assemble the enchiladas, fill each tortilla with shredded birria meat, a sprinkle of diced onion, and a little cheese. Roll them up and place them seam-side down in a greased baking dish.
Pour the reserved birria broth over the top of the enchiladas and sprinkle with additional cheese.
Bake in the preheated oven for about 20 minutes until the cheese is melted and bubbly.
Remove from the oven and garnish with chopped cilantro.
Serve with lime wedges on the side for squeezing over the enchiladas.