Step 1Preheat oven to 350º. In a large, heavy pot or Dutch oven over medium heat, toast guajillo, morita, and pasilla chiles, stirring frequently, until softened and fragrant, 2 to 3 minutes.
Step 2Transfer chiles to a medium heatproof bowl, then cover with boiling water. Using a small plate or bowl, completely submerge chiles. Let soak until rehydrated and pliable, about 20 minutes.
Step 3Meanwhile, generously season brisket and oxtails with salt and pepper. In same pot over medium-high heat, heat oil. Working in batches, cook beef, turning occasionally, until browned on all sides, 6 to 7 minutes per side for brisket/roast, 4 to 5 minutes for bone-in parts. Transfer to a plate or cutting board.
Step 4In a blender, blend chiles, 1 1/2 cups chile liquid, tomatoes, garlic, cloves, cinnamon, vinegar, cumin, and oregano, adding more chile liquid if needed, until a pourable paste forms, about 1 minute.
Step 5Return beef to pot and add onion, bay leaves, and chile paste. Pour in enough water to just cover beef; season with a heavy pinch of salt. Bring to a simmer over medium heat.
Step 6Remove from heat, cover, and bake until beef is fork-tender, 4 to 4 1/2 hours. Discard bay leaves and onion, transfer beef to a cutting board, and reserve broth (or consomé). Shred beef with 2 forks.
Step 7Taste consomé; season with salt and pepper. Depending on your preference, to thin out consomé, add water or chicken or beef stock until desired consistency is reached.
Step 8Make Ahead: Birria can be made 5 days ahead. Store beef and consomé in separate airtight containers and refrigerate.
ASSEMBLY
Step 1In same large pot over medium-low heat, bring consomé to a bare simmer (there should be a layer of dark red-looking fat/oil on top).
Step 2Construct a taco station: Have the consomé simmering on one burner, a large cast-iron skillet or plancha heated over medium heat on another, a plate of shredded beef from the birria and another plate of tortillas to the side, as well as small bowls filled with Oaxaca cheese, onions, and cilantro.
Step 3Working in batches, reheat 1/3 cup beef on one side of cast-iron skillet, leaving enough room for a tortilla, tossing occasionally, until seared and browned. Take a tortilla and dip into consomé, coating both sides with dark red fat. Place tortilla on cast-iron skillet next to beef and cover with cheese.
Step 4Fry tortilla until cheese is mostly melted and underside has browned and started to crisp, about 3 minutes. Place seared meat on one half of tortilla and top with onion and cilantro. Fold tortilla into a taco and cook until crisp, about 30 seconds per side. Repeat with remaining beef.
Step 5Serve tacos with lime wedges and small bowls of consomé for dipping alongside.
