01 - Toast the guajillo, ancho, and pasilla chiles in a dry skillet over medium heat for 1 to 2 minutes, turning frequently, until fragrant and slightly pliable. Transfer to a heatproof bowl, cover with hot water, and soak for 15 minutes until fully softened.
02 - Heat the oil in a large Dutch oven or heavy-bottomed pot over medium-high heat. Working in batches, brown the beef chuck and short ribs on all sides until a deep golden crust forms, approximately 3 to 4 minutes per side. Remove the meat and set aside.
03 - In the same pot, sauté the quartered onion, garlic cloves, and Roma tomatoes over medium heat until softened and lightly charred, about 5 minutes.
04 - Drain the soaked chiles and transfer them to a blender. Add the sautéed onion, garlic, and tomatoes along with the cinnamon stick, black peppercorns, cumin seeds, coriander seeds, whole cloves, dried oregano, dried thyme, apple cider vinegar, and 1 cup of the beef broth. Blend on high until completely smooth.
05 - Pour the blended sauce through a fine-mesh strainer back into the pot, pressing with a spatula to extract all the liquid. Return the seared beef to the pot, then pour in the remaining 3 cups of beef broth. Add the bay leaves and salt, stirring to combine. Cover, reduce heat to low, and simmer gently for approximately 3 hours, until the beef is fall-apart tender and easily shreds with a fork.
06 - Remove the beef from the pot and shred it finely using two forks, discarding any large pieces of fat or bone. Skim the rendered fat from the surface of the consommé and reserve it in a separate bowl for frying the tortillas. Keep the consommé warm for serving.
07 - Heat a skillet or griddle over medium-high heat. Brush one side of each corn tortilla with the reserved birria fat. Place tortillas fat-side down on the hot surface. Add a generous mound of shredded beef and a portion of cheese onto one half of each tortilla. Fold in half to form tacos and cook until the exterior is golden and crisp, about 2 to 3 minutes per side.
08 - Arrange the crispy tacos on a platter and garnish generously with the diced white onion and chopped fresh cilantro. Serve immediately with lime wedges alongside small bowls of warm birria consommé for dipping.