Step 1: Rehydrate Chiles: Place guajillo and ancho chiles in a heatproof bowl. Cover with boiling water and let soak for 20-30 minutes until soft. Drain, reserving a bit of the soaking liquid if needed, then remove stems and seeds.
Step 2: Prepare Chili Paste: In a blender, combine the rehydrated chiles, 1/4 of the chopped white onion, 3 cloves of garlic, cumin, oregano, and 1 cup of beef broth. Blend until completely smooth. For an extra smooth consomé, strain this paste through a fine-mesh sieve into a bowl, pressing on the solids.
Step 3: Sear Beef: Pat beef chuck roast dry and season generously with salt and pepper. Heat olive oil in a large Dutch oven or heavy pot over medium-high heat. Sear beef chunks on all sides until deeply browned, working in batches if necessary. Remove beef and set aside.
Step 4: Simmer Birria: Return all seared beef to the pot. Pour the chili paste over the beef. Add the remaining 3 cups of beef broth, apple cider vinegar, and bay leaves. Bring to a gentle simmer.
Step 5: Slow Cook: Reduce heat to low, cover tightly, and simmer for 2.5 to 3 hours, or until beef is fork-tender. (Alternatively, transfer to a slow cooker and cook on low for 6-8 hours or high for 3-4 hours).
Step 6: Shred Beef & Skim Consomé: Remove beef from the pot and shred it into bite-sized pieces using two forks. Discard bay leaves. Skim off any excess fat from the surface of the consomé.
Step 7: Prepare Tortillas: Heat a large skillet or comal over medium heat. Dip each corn tortilla briefly into the warm consomé, coating both sides. Place the dipped tortilla onto the hot skillet.
Step 8: Assemble Tacos: If making quesabirria, sprinkle about 1/4 cup of shredded cheese on one half of the tortilla on the skillet. Add a generous portion of shredded birria meat. Cook until the tortilla is slightly crispy and the cheese is melted (if using). Fold the tortilla in half to form a taco. Repeat with remaining tortillas and meat.
Step 9: Serve: Serve the birria tacos immediately with small bowls of the warm consomé for dipping. Garnish with fresh chopped cilantro, diced white onion, and lime wedges.