In a large pot, heat the vegetable oil over medium-high heat. Add the beef chunks and shank, browning them on all sides for about 5-7 minutes. Remove the meat and set aside.
In the same pot, add the onion and garlic, sautéing until fragrant and translucent, about 3-4 minutes.
While the onion and garlic are cooking, prepare the chiles. In a small saucepan, add the guajillo and ancho chiles with enough water to cover them. Bring to a boil, then reduce heat and simmer for about 10 minutes until softened. Drain and transfer to a blender.
Add the cumin, oregano, cinnamon, black pepper, salt, beef broth, and apple cider vinegar to the blender with the softened chiles. Blend until smooth.
Return the browned meat to the pot and pour the blended sauce over it. Stir to combine, then bring to a simmer. Cover and reduce heat to low, cooking for 2-3 hours until the meat is tender and easily shredded.
Once cooked, remove the meat from the pot and shred it using two forks. Return the shredded meat to the pot, stirring to combine with the sauce.
Serve the birria hot, garnished with fresh cilantro and lime wedges. Pair with warm corn tortillas for dipping.