01 - Toast dried guajillo, ancho, and pasilla chiles in a dry skillet over medium heat for 1-2 minutes until fragrant. Soak in hot water for 10 minutes until soft, then drain thoroughly.
02 - Combine softened chiles, onion, garlic, diced tomatoes, vinegar, oregano, cumin, paprika, cinnamon, cloves, salt, and black pepper in a blender. Puree until completely smooth, scraping down sides as needed.
03 - Place beef chuck and short ribs in a Dutch oven or heavy pot. Pour blended sauce over meat. Add beef broth and bay leaf. Mix thoroughly to coat all pieces. Bring to a simmer over medium heat.
04 - Cover pot and reduce heat to low. Cook for 3 to 3.5 hours until meat is fork-tender and easily shreds. Skim excess fat from surface and reserve for frying tortillas.
05 - Remove beef from cooking liquid. Shred meat using two forks, discarding bones and bay leaf. Return shredded meat to pot and stir well to coat in consommé.
06 - Arrange chopped cilantro, diced onion, and lime wedges in separate serving bowls.
07 - Heat skillet over medium heat with reserved fat or oil. Dip tortilla in birria consommé to coat lightly. Place on skillet, sprinkle cheese over half, add meat, and fold tortilla. Cook 2-3 minutes per side until crispy and golden. Repeat with remaining tortillas.
08 - Ladle consommé into small bowls for dipping. Serve tacos garnished with onion, cilantro, and lime wedges alongside consommé.