Carrots pull double duty here -- they go into the poaching liquid for the chicken, and they're tossed with rice vinegar to make a crunchy topping for the lettuce cups.
Combine chicken, 4 cups water, 2 tablespoons salt, 1 tablespoon ginger, scallion tops, and chopped carrot in a medium saucepan (if chicken is not entirely covered, add more water). Bring to a boil, then reduce heat and partially cover; simmer until a thermometer inserted in thickest parts of chicken (without touching bones) reaches 165 degrees, 15 to 20 minutes. Remove chicken; let cool slightly. Remove meat from bones; shred into large pieces (discard skin and bones).
Meanwhile, combine sliced scallions, remaining 3 tablespoons ginger, and oil in a bowl. Season with salt and pepper. Toss half of scallion sauce with chicken; season with salt and pepper. Toss carrot ribbons with vinegar. Just before serving, place chicken in lettuce cups; top with peanuts, cilantro, carrots, and remainingscallion sauce.
Use a Y-peeler to cut the carrots into ribbons for pickling. Alternatively use a box grater; hold the carrots parallel to the holes of the grater for longer strands.