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 remaining scallion sauce.
