Put the chicken in a medium saucepan and add 2 to 3 cups of water, enough to cover the chicken by about ½ inch. Bring to a rolling boil over medium-high heat, reduce the heat to maintain a lively simmer, and cook until done, 10 to 15 minutes. Meanwhile, combine the lime juice, fish sauce, vinegar, sugar, and pepper in a medium bowl, and stir to dissolve the sugar and mix everything well. Add the onion and toss to coat. Set aside for 20 to 30 minutes, until you are ready to complete the dish.

Transfer the meat to a plate to cool, reserving the broth for another use, such as making soup or cooking rice. When the chicken is cool, tear it into long, thin shreds. Coarsely chop the mint and the rau ram, if you are using it. Add the shredded chicken, cabbage, carrots, mint, and rau ram to the bowl of onions and seasonings and toss to coat everything well. Mound the salad on a serving plate and top with chopped peanuts, if you are using them. Serve at room temperature or chilled.