Prepare the broth: Fill a large pot with chicken broth, beef broth, or lightly salted water. Add your choice of vegetables and aromatic herbs. A simple aromatic base consists of 1–2 onions, a couple of bay leaves, whole peppercorns, and a carrot. Feel free to include other herbs and spices to your taste, such as oregano, rosemary, garlic, or chili pepper. Bring the liquid to a rolling boil over high heat.
Add the tongue: Place the meat into the boiling broth and cover the pot with a lid. Wait for the liquid to return to a boil, then reduce the heat to maintain a gentle simmer. Keep the meat fully submerged throughout cooking; you may need to add more liquid or weigh the tongue down with a steamer basket.
Simmer until tender: Allow the tongue to simmer until completely tender. It is cooked when the outer skin turns pale/white and a knife easily pierces through the thickest section. Generally, this takes about 50 to 60 minutes per pound (roughly 500 g) of meat. If cooked too quickly or undercooked, the meat will be tough and unpleasant to eat; if you have the time, it is safer to cook it an hour or two longer.
Peel while warm: Transfer the tongue to a platter using kitchen tongs. Let it sit just until it is cool enough to handle, then use a sharp knife to score the outer white skin lengthwise. Peel the outer skin layer away with your fingers, trimming with the knife as needed. (While technically edible, the skin has an unappealing texture and flavor). Tongue becomes significantly harder to peel as it cools; if it cools to room temperature, dipping it in ice water can help loosen the skin. Reserve the flavorful cooking broth for making soup or enriching sauces.
Slice and serve: Cut the meat into 1/4-inch (6 mm) thick slices, cutting on a diagonal with a sharp knife. Serve with Mexican salsa verde, alongside crusty bread with spicy mustard and leafy greens, or roast in the oven for an additional 30 minutes with roasted potatoes. Since this yields a generous amount of meat, you can reserve larger pieces to sear on the grill or incorporate into subsequent recipes.
Note: If the meat feels tough, it is simply undercooked—return it to the broth and continue simmering. You can also turn part of the broth into gravy by thickening it with a little flour.