In a large Dutch oven, heat the vegetable oil over medium-high heat. Add the beef chuck roast and beef shank, browning them on all sides for about 5-7 minutes. This step adds depth of flavor through caramelization.

Once browned, add the quartered onion, minced garlic, and water to the pot. Ensure the meat is submerged.

In a separate pot, bring water to a boil and add the guajillo and ancho chiles. Boil for about 10 minutes until softened. This process rehydrates the chiles, enhancing their flavor.

Transfer the softened chiles to a blender, adding a cup of the boiling water and blend until smooth. This creates a rich sauce.

Pour the blended chile sauce into the pot with the meat. Add cumin, oregano, black pepper, salt, and bay leaf. Stir to combine.

Bring the mixture to a boil, then reduce the heat to low. Cover and simmer for 3-4 hours, or until the meat is tender and easily shredded. Stir occasionally and check the liquid level, adding more water if necessary.

Once the meat is tender, remove it from the pot and shred it with two forks. Discard any large pieces of fat or bone.

Strain the broth through a fine-mesh strainer into a bowl, discarding solids. This broth will be used for dipping.

To assemble the tacos, take a warm tortilla, fill it with shredded birria, and top with chopped cilantro, onion, and crumbled queso fresco.

Serve the tacos with a side of the strained broth for dipping, and lime wedges for squeezing over the top.