Remove the stems and seeds from the guajillo and ancho chilies. Soak them in hot water for about 15 minutes until softened.

In a blender, combine the soaked chilies, chopped onion, minced garlic, cumin, oregano, and a half cup of beef broth. Blend until smooth.

Heat a large pot over medium-high heat and add a drizzle of oil. Add the beef chuck pieces, season with salt and pepper, and cook until browned on all sides, about 5-7 minutes.

Pour the chili mixture over the browned beef and add the remaining beef broth. Bring to a boil, then reduce to a simmer. Cover and cook for about 2 hours, or until the meat is fork-tender.

Remove the beef from the pot and shred it with two forks. Return the shredded meat to the pot to soak up the flavors.

In a separate pan, lightly toast corn tortillas. Fill each tortilla with shredded beef and cheese, then fold and cook on a skillet until cheese is melted and tortillas are crispy.

Serve the birria with a bowl of the broth on the side for dipping, garnished with chopped cilantro and lime wedges.