01 - Toast the dried guajillo, ancho, and pasilla chiles in a dry skillet over medium heat for 1-2 minutes until fragrant. Soak them in hot water for 10 minutes until soft, then drain well.
02 - Combine softened chiles, onion, garlic, tomatoes, vinegar, oregano, cumin, paprika, cinnamon, cloves, salt, and black pepper in a blender. Blend until completely smooth to create the birria sauce.
03 - Place beef chunks and short ribs in a large Dutch oven or heavy pot. Pour the blended sauce over the meat. Add beef broth and bay leaf. Mix thoroughly to coat all pieces. Bring to a simmer over medium heat, then cover and reduce to low. Cook for 3–3.5 hours until meat is fork-tender and shreds easily. Skim fat from the top and reserve for frying.
04 - Remove the beef from the pot and shred using two forks. Discard bones and bay leaf. Return shredded meat to the pot and stir well to coat in the consommé.
05 - Arrange chopped cilantro, diced onion, and lime wedges in separate serving bowls.
06 - Heat a skillet over medium heat with a bit of reserved fat or oil. Dip a tortilla into the top layer of birria consommé to coat in flavored fat. Place on the skillet, sprinkle cheese over half, then top with birria meat. Fold tortilla closed. Cook 2–3 minutes per side, pressing gently, until crispy and golden. Repeat with remaining tortillas.
07 - Ladle consommé into small bowls for dipping. Serve tacos garnished with chopped onion, cilantro, and lime wedges alongside the consommé.