Toast the Chilies: In a dry skillet over medium heat, toast the guajillo, ancho, and pasilla chilies for 1–2 minutes until fragrant.
Soak the Chilies: Transfer to a bowl of hot water and soak for 15–20 minutes until softened.
Make the Sauce: In a blender, combine soaked chilies, tomatoes, garlic, onion, vinegar, cumin, cinnamon, paprika, and 1 cup of beef broth. Blend until smooth and strain if desired.
Sear the Beef: Heat oil in a skillet. Sear seasoned beef on all sides until browned.
Cook the Birria: In a Dutch oven or slow cooker, add seared beef, blended sauce, remaining broth, bay leaves, oregano, salt, and pepper. Simmer on low for 4 hours (or 8 hours on low in a slow cooker).
Shred the Meat: Remove the beef, shred it with forks, and return it to the consommé to soak.
Assemble Tacos: Dip tortillas in the top layer of consommé, fry in a skillet, add cheese and beef, fold, and crisp on both sides.
Serve: Garnish with chopped onion, cilantro, and lime. Serve with a small bowl of consommé for dipping.