01 - Heat a dry skillet over medium heat. Toast guajillo, ancho, and pasilla chiles for 1–2 minutes until fragrant. Transfer to a bowl, cover with hot water, and 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 - Combine softened chiles, roasted vegetables, vinegar, oregano, thyme, cumin seeds, peppercorns, cloves, and 1 cup beef broth in a blender. Purée until completely smooth.
04 - Heat vegetable oil in a large Dutch oven over medium-high heat. Season beef chuck and short ribs with salt. Sear on all sides until deeply browned.
05 - Pour blended sauce over browned beef. Add remaining 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.
06 - Remove beef from pot. Shred using two forks, discarding bones and excess fat. Skim fat from the consommé, reserving a small amount for frying tortillas.
07 - Preheat oven to 375°F. Dip each corn tortilla in warm consommé, then fry briefly in a hot skillet with reserved fat until pliable, 10–15 seconds per side.
08 - Fill each tortilla with shredded birria and a sprinkle of cheese. Roll tightly and place seam-side down in a baking dish.
09 - Pour remaining consommé over enchiladas and top with remaining cheese. Bake for 10–15 minutes until cheese is melted and bubbly.
10 - Garnish hot enchiladas with diced onion, chopped cilantro, and lime wedges.