Prepare the Chiles: Toast the dried chiles in a dry skillet over medium heat for 2-3 minutes per side, until fragrant and slightly pliable. Be careful not to burn them. Rehydrate the toasted chiles by soaking them in hot water for 20-30 minutes, until softened.

Blend the Chile Paste: Drain the rehydrated chiles and transfer them to a blender. Add 1 cup of the soaking water, onion, garlic, oregano, cumin, cinnamon, cloves, apple cider vinegar, salt, and pepper. Blend until smooth. If needed, add more soaking water to help the blending process.

Sear the Beef: Season the beef chuck roast and beef shank generously with salt and pepper. Heat a large Dutch oven or heavy-bottomed pot over medium-high heat. Sear the beef on all sides until browned, about 3-4 minutes per side. This is crucial for developing flavor.

Combine and Simmer: Pour the chile paste over the seared beef. Add the beef broth, water, and bay leaf. Bring to a simmer, then reduce the heat to low, cover, and cook for 3-4 hours, or until the beef is incredibly tender and easily shreds with a fork.

Shred the Beef: Remove the beef from the pot and shred it using two forks. Discard the bay leaf.

Strain the Broth: Strain the broth through a fine-mesh sieve, pressing down on the solids to extract as much liquid as possible. This creates the consommé for dipping the tacos.

Assemble the Tacos: Dip a corn tortilla into the top layer of fat that has accumulated on the strained broth. Place the tortilla on a hot griddle or skillet. Add shredded cheese and shredded beef to one half of the tortilla. Fold the tortilla in half and cook until the cheese is melted and the tortilla is crispy and golden brown, about 2-3 minutes per side.

Serve: Serve the birria tacos immediately with chopped cilantro, chopped onion, lime wedges, and a small bowl of the consommé for dipping.