Combine beef, onion, garlic, bay leaves, cumin, oregano, chili powder, cinnamon, salt, and pepper in a large pot. Cover with beef broth or water.
Bring to a boil, then reduce heat and simmer for 2.5-3 hours until tender.
Remove the meat, shred it, and strain the consommé for dipping.
Heat a skillet over medium. Dip tortillas in consommé, place on the skillet, and sprinkle cheese on one half.
Add shredded birria on top of the cheese, fold the tortilla, and press gently.
Cook for 2-3 minutes per side until crispy and cheese melts.
Serve hot with chopped cilantro, diced tomatoes, onions, lime wedges, and consommé for dipping.