Place the meat in a large saucepan or Dutch oven with the onion, 2 of the garlic cloves, 6 sprigs of mint, the peppercorns and 1 teaspoon salt (or to taste). Add enough cold water to cover by 1 inch. Bring to a boil over high heat; quickly reduce the heat to medium-low and cook, partly covered, until the meat is well cooked, about 1½ to 2 hours. Remove the meat, saving the stock for another purpose.

When the meat is cool enough to handle, cut it across the grain into 3 large chunks and pull into fine shreds. Transfer the meat to a bowl large enough to hold all the remaining ingredients.

Make a vinaigrette: Peel the remaining garlic clove and using a mortar and pestle or the flat of a heavy knife blade, crush it to a paste with the remaining 1 teaspoon salt. In a small bowl, whisk together the crushed garlic with the vinegar, olive oil, and freshly ground pepper to taste. Pour the dressing over the shredded meat and toss to combine. Let marinate, covered, for 1 hour at room temperature.

Prepare the other salad ingredients: Finely chop the tomatoes. Remove and discard the tops and seeds of the chiles; mince the flesh. Slice the radishes thinly and cut into fine julienne. Peel and pit the avocados; cut the flesh into ½-inch dice. Strip the leaves from the remaining mint sprigs and chop coarsely. Add all these ingredients to the marinated meat and toss to combine thoroughly. Taste for seasoning; add more salt and pepper if desired.