Prepare chiles: Toast dried chiles in a dry skillet for 1–2 minutes until fragrant. Then soak in hot water for 10 minutes.
Blend sauce: In a blender, combine soaked chiles, tomato, garlic, onion, vinegar, and spices (except bay leaves and cinnamon). Blend until smooth.
Sear meat: Season beef with salt and pepper. In a large pot, sear meat on all sides in oil until browned.
Simmer: Add blended sauce, beef broth, bay leaves, and cinnamon stick to the pot. Bring to a boil, reduce heat, cover, and simmer for 2.5–3 hours until meat is tender and shreds easily.
Shred beef: Remove meat, shred it with forks, and return it to the consommé to keep moist.
Assemble tacos: Heat a skillet over medium heat. Dip tortillas in the top layer of consommé, place on the skillet, and top with cheese and shredded birria. Fold and cook until crispy on both sides.
Serve hot with a cup of warm consommé for dipping, chopped onion, cilantro, and lime wedges.