Prep the Meat: Ensure the beef tongue and cheek are thoroughly cleaned, removing any excess fat or tough bits.
Make the Marinade: Blend half an onion, garlic cloves, allspice, peppercorn, kosher salt, and oregano until smooth.
Marinate the Meat: Place the cleaned meat in a Dutch oven or oven-safe pan. Pour the blended marinade over the meat. Add the remaining half onion pierced with cloves and bay leaves. Cover tightly with foil and a lid to trap steam.
Cook: Bake in a preheated oven at 350°F (175°C) for 3.5 hours.
Finish: Carefully remove from the oven and transfer the meat to a cutting board. Peel the top layer of skin from the beef tongue and roughly chop the meat.
Serve: Serve the chopped meat in warm tortillas. Garnish with cilantro and serve with lime wedges and salsa verde
