Bring the grill up to 225 degrees F (105 degrees C).
Fill a Dutch oven two-thirds full with water. Add the tongue, onions, crushed garlic cloves, bay leaves, peppercorns, and salt. Bring to a boil, reduce the temp on the grill to 200 degrees F (90 degrees C) or to a simmer, cover, and cook for 3 hours, until the tongue is soft to the touch and tender.
Remove the tongue from the water and let cool for a couple of minutes.
Using your fingers and/or a sharp small knife, remove the light-colored skin-like covering from the tongue and discard. There is a rough patch of meat where the tongue attaches to the bottom of the mouth. It’s a little rough, but perfectly edible. Keep or remove, as desired. Slice the tongue into 1/4-inch-thick slices. (If you are not preparing the whole tongue for tacos, you can return whatever tongue you do not slice to the cooking water to soak.)
Place a cooking surface (soapstone, cast iron griddle, etc.) on the grill and heat to 450 degrees F (230 degrees C).
Heat a little oil on the cooking surface and sauté the slices on both sides until they are lightly browned. Remove from the cooking surface. Slice into strips, and then crosswise again so that you end up with small cubes.
Soften tortillas either by cooking on the hot grill surface until pockets of air appear in them, or in the microwave (about 10 seconds per tortilla).
Place a large spoonful of meat in the center of a tortilla. Add a spoonful of salsa verde and some chopped avocado, onions, and cilantro. Garnish with radish slices and a drizzle of lime juice. For extra spice, add jalapeño peppers.