Step 1: Toast the dried guajillo, ancho, and arbol chilies in a dry skillet over medium heat for 2-3 minutes until fragrant. Remove stems and seeds. Place in a heatproof bowl and cover with boiling water; let rehydrate for 20-30 minutes.
Step 2: While chilies rehydrate, lightly char the roughly chopped white onion and peeled garlic cloves in a dry skillet or under a broiler until softened and slightly blackened.
Step 3: Drain the rehydrated chilies, reserving 1 cup of the soaking liquid. In a blender, combine softened chilies, charred onion and garlic, diced tomatoes, apple cider vinegar, cumin, Mexican oregano, cloves, cinnamon stick, bay leaves, peppercorns, and salt. Add 1/2 cup of reserved chili liquid. Blend until smooth, adding more liquid if needed. Strain the adobo through a fine-mesh sieve, pressing solids to extract liquid; discard solids.
Step 4: Pat beef chuck roast and short ribs dry; season generously with salt and pepper. Heat 2 tbsp olive oil in a large Dutch oven over medium-high heat. Sear beef in batches until deeply browned on all sides. Remove beef and set aside.
Step 5: Add the strained adobo paste to the Dutch oven, stirring and scraping up any browned bits. Simmer for 5 minutes. Return seared beef to the pot, ensuring it’s mostly submerged. Add beef broth to just cover the meat. Bring to a simmer, then reduce heat to low, cover, and cook for 3-4 hours, or until beef is fall-apart tender.
Step 6: Remove cooked beef from the pot; shred with two forks, discarding large pieces of fat. Skim any excess fat from the braising liquid (consommé) and adjust seasoning. Keep shredded birria and consommé warm.
Step 7: Heat a large cast-iron skillet or griddle over medium-high heat. Dip a corn tortilla into the consommé, coating both sides. Place on the hot skillet. Sprinkle with shredded Oaxaca cheese. Add shredded birria to one half of the tortilla.
Step 8: Cook for 1-2 minutes until cheese melts and tortilla is lightly crispy. Fold the other half of the tortilla over to form a taco. Press down gently. Cook for another 1-2 minutes on each side until golden brown and crispy. Repeat with remaining tortillas and filling.
Step 9: Serve hot birria tacos immediately with small bowls of consommé for dipping. Garnish with finely diced white onion, fresh cilantro, and lime wedges.