Start by toasting your dried chiles in a dry skillet over medium heat for about 30 seconds per side, just until they're fragrant. Be careful not to let them burn or they'll turn bitter. Once toasted, place them in a bowl and cover with boiling water. Let them soak for about 15 to 20 minutes until softened, then drain and set aside.
While the chiles soak, heat vegetable oil in a large Dutch oven or heavy-bottomed pot over medium-high heat. Season your beef chuck and short ribs generously with salt and pepper. Sear the meat in batches for about 3 to 4 minutes per side until a deep brown crust forms. Don't rush this step — that crust is where so much of the flavor lives. Remove the meat and set it aside.
In the same pot, add the halved onion cut-side down and the garlic cloves. Let them char slightly for about 2 minutes. This adds a beautiful smokiness to the broth. Then add the diced tomatoes and cook for another 2 minutes, scraping up any browned bits from the bottom of the pot.
Transfer the softened chiles, charred onion, garlic, and tomatoes to a blender. Add 1 cup of beef broth along with the cumin, oregano, smoked paprika, cinnamon, cloves, and apple cider vinegar. Blend everything on high until completely smooth. If the blender struggles, add a splash more broth. Strain the blended sauce through a fine mesh strainer back into the pot for an extra smooth consommé.
Return the seared beef to the pot and pour in the remaining 3 cups of beef broth and 1 cup of water. Add the bay leaves and give everything a good stir. Bring it to a boil, then reduce the heat to low, cover, and let it braise gently for about 3 hours. You're looking for meat that shreds effortlessly when you touch it with two forks.
Once the meat is done, carefully remove it from the pot and shred it using two forks, discarding any large bones and excess fat. Taste the consommé and adjust the salt as needed. Keep it warm on the stove — this liquid is liquid gold and you'll be dipping your tacos into it.
Heat a large skillet or griddle over medium-high heat. Dip a corn tortilla into the top layer of the consommé (where the fat floats) to lightly coat both sides. Place it in the skillet, add a handful of shredded cheese to one half, then pile on a generous scoop of shredded birria meat. Fold the tortilla over and cook for about 2 to 3 minutes per side until crispy and golden with melted cheese inside. Repeat with remaining tortillas.
Serve the tacos immediately with small cups of warm consommé for dipping, topped with diced onion, fresh cilantro, and a big squeeze of lime.