Brown the beef: Pat the beef dry and season with kosher salt and black pepper. Heat the oil in a large Dutch oven over medium-high heat. Working in batches, sear the beef on all sides until deeply browned. Transfer to a plate. Don't overcrowd the pot.
Toast the chiles: Heat a dry skillet over medium heat. Toast the guajillo, ancho, and árbol chiles for about 15–30 seconds per side, turning frequently. They should smell fragrant but should not blacken. Transfer to a bowl and cover with hot water. Soak for about 15 minutes.
Blend the chile sauce: Drain the softened chiles and add them to a blender with the garlic, onion, tomatoes, apple cider vinegar, Mexican oregano, cumin, cinnamon, cloves, and 1 cup of the beef broth. Blend until smooth. Strain through a fine-mesh sieve if you want an especially smooth consommé.
Braise the beef: Return the browned beef to the Dutch oven. Pour the chile mixture around the meat and add the remaining 3 cups of beef broth. Add the bay leaves. The liquid should come roughly halfway up the beef. Bring to a gentle simmer.
Slow-cook: Cover the Dutch oven and transfer to a 325°F oven. Cook for 3–4 hours, checking toward the end, until the beef is extremely tender and pulls apart easily with a fork. The exact time will depend on the size of the beef pieces and your particular cut.
Shred and prepare the consommé: Remove the beef and shred it with two forks. Discard large pieces of connective tissue or excess fat. Strain the braising liquid if desired. Skim the flavorful beef fat from the surface and reserve it for frying the tortillas. Keep the beef and consommé hot.
Crisp the tacos: Heat a large skillet over medium-high heat. Lightly dip one side of a corn tortilla into the reserved beef fat and place it fat-side down in the skillet. Add shredded cheese to one half, followed by shredded birria beef. Fold the tortilla over and cook until deeply golden and crisp. Flip once and cook the other side until crisp and the cheese is melted. Repeat with the remaining tacos.
Serve: Pour hot consommé into small bowls. Top with diced white onion, cilantro, and a squeeze of lime. Serve the crispy tacos immediately alongside the consommé for dipping.