01 - Place guajillo, ancho, and pasilla chiles in a dry skillet over medium heat. Toast for 1 to 2 minutes, turning occasionally, until fragrant and pliable. Transfer to a heatproof bowl, cover with hot water, and soak for 15 minutes until fully softened.
02 - Heat oil in a large Dutch oven or heavy pot over medium-high heat. Working in batches, brown beef chuck and short ribs on all sides until a deep crust forms, about 3 to 4 minutes per batch. Remove meat and set aside.
03 - In the same pot, add the quartered onion, garlic cloves, and quartered Roma tomatoes. Sauté over medium heat until softened and lightly charred, about 5 minutes.
04 - Drain the soaked chiles and transfer to a blender. Add the sautéed onion, garlic, and tomatoes along with cinnamon stick, black peppercorns, cumin seeds, coriander seeds, whole cloves, dried oregano, dried thyme, apple cider vinegar, and 1 cup of 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 liquid. Return the seared beef to the pot and add the remaining 3 cups of beef broth, bay leaves, and salt. Stir to combine, cover, and simmer gently over low heat for approximately 3 hours, until the beef is fall-apart tender and easily shredded with a fork. Skim the rendered fat from the surface and reserve it for frying the tacos.
06 - Remove the beef from the broth and shred using two forks, discarding any large pieces of fat or bone. Strain the remaining braising liquid and reserve as consommé for dipping.
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 portion of shredded beef and a sprinkle of cheese to one half of each tortilla. Fold in half to form tacos and cook until golden and crispy, about 2 to 3 minutes per side.
08 - Arrange crispy birria tacos on a platter and top with finely diced white onion and chopped fresh cilantro. Serve immediately with lime wedges and small bowls of warm birria consommé for dipping.