Place the meat in a 6-quart Dutch oven with the onion, garlic, peppercorns, and salt. Add 1 quart water, bring to a boil, and adjust the heat to low. Cook, covered, for 1 to 1½ hours, or until the meat is fork-tender. Remove from the heat and let cool to lukewarm in the cooking broth.

While the meat cooks and cools, peel the red onion and cut into medium-fine dice. Trim the radishes and slice into thin rounds. Prepare the chosen dressing.

Remove the cooled meat from the pot, letting it drain well. (Strain the stock and save it for a soup or sauce.) Place it on a cutting board and pull into fine shreds, using your fingers or two forks.

Pile the shredded meat onto a serving platter. Add the red onion and radishes, reserving a little of each for garnish. Add ½ cup of the dressing and toss to mix everything thoroughly. Add a little more of the dressing if needed. Garnish with cilantro and the reserved onion and radish. Just at serving time, pit and peel the avocado, cut it lengthwise into wedges, and arrange them over the salad.