01 - In a dry skillet over medium heat, toast the dried guajillo, ancho, and pasilla chiles for 1–2 minutes until fragrant. Transfer to a bowl and cover with hot water; soak for 15 minutes until softened.
02 - In the same skillet, roast quartered onion, garlic cloves, and halved tomatoes until slightly charred, about 5 minutes.
03 - Drain the soaked chiles and transfer to a blender. Add roasted vegetables, vinegar, oregano, thyme, cumin seeds, peppercorns, cloves, and 1 cup beef broth. Blend until completely smooth.
04 - In a large Dutch oven, heat vegetable oil over medium-high heat. Sear beef chuck chunks and short ribs on all sides until well browned, about 3–4 minutes per batch.
05 - Pour the blended sauce over the beef. Add remaining 3 cups beef broth, bay leaves, cinnamon stick, and 2 tsp salt. Bring to a boil, then reduce heat to low. Cover and simmer for 2.5–3 hours until meat is fork-tender and easily shreds.
06 - Remove beef from the pot and shred with two forks, discarding bones and excess fat. Skim excess fat from the birria consommé and reserve 2–3 tablespoons for frying tortillas.
07 - Preheat oven to 375°F. Dip each corn tortilla in the warm birria consommé, then fry briefly in a hot skillet with a little reserved fat until pliable, about 10–15 seconds per side.
08 - Fill each tortilla with shredded birria and a sprinkle of cheese. Roll up and place seam-side down in a baking dish.
09 - Pour extra birria consommé over the enchiladas and top with remaining cheese. Bake for 10–15 minutes until cheese is melted and bubbly.
10 - Serve hot, garnished with diced onion, chopped cilantro, and lime wedges on the side.