Place the pigs' feet, water, 3 cups of the red wine vinegar, the black peppercorns, bay leaves, 1 tablespoon of the salt and the dried thyme in a large stockpot. Bring to a boil, reduce to a simmer and cook, uncovered, until the skin starts to pull away from the bones and the meat slips off easily when pierced with a fork, 1 1/2 to 2 hours. Set aside to cool completely, then refrigerate 1 hour.
Remove and discard the outer layer of skin and fat from the pigs' feet and discard the bones. Slice the meat and soft cartilage into 1 x 1/8-inch thick strips.
Place the meat, carrot and all the peppers in a large mixing bowl. Add the olive oil, the remaining 1/2 cup red wine vinegar, the 3/4 teaspoon salt and the pepper and toss well to combine. Cover and chill 2 to 4 hours. Serve cold on lettuce-lined plates.