In a large pot or Dutch oven over medium heat, add the chunks of beef and brown them on all sides, about 10 minutes.
Soak the guajillo and ancho chilies in hot water for about 15 minutes until softened. Drain and place them in a blender or food processor with the onion, garlic, beef broth, cumin, and oregano. Blend until smooth.
Pour the chili sauce over the browned meat. Add enough water to cover the meat if needed. Season with salt and pepper. Bring to a simmer, cover, and cook on low heat for about 60 minutes, until the meat is tender and can be easily shredded.
Preheat the oven to 350°F (175°C). Shred the cooked meat using two forks and reserve some of the cooking liquid for later.
In a frying pan, lightly toast the corn tortillas on both sides until pliable.
Fill each tortilla with shredded birria meat and sprinkle a generous amount of cheese on top. Roll the tortillas tightly and place them seam side down in a baking dish.
Pour the reserved cooking liquid over the filled tortillas and sprinkle with any remaining cheese. Bake in the preheated oven for 15-20 minutes, or until the cheese is melted and bubbly.
Remove from the oven and let cool slightly. Serve garnished with chopped cilantro and diced onion, with lime wedges on the side.