Pat the chicken dry with paper towels. Pound it to an even thinness.
Place the chicken in a large saucepan and add water to cover by ½ inch. Add 3 teaspoons salt and ½ teaspoon pepper and bring to a gentle simmer.
Cook until no trace of pink remains, 8 to 10 minutes. Transfer to a bowl of ice water for 5 minutes.
Meanwhile, in a large bowl, combine the lime juice, vinegar, and sugar, stirring until the sugar dissolves. Add the scallions and apples and toss.
Drain, pat dry, and dice the chicken. Add it to the apple mixture with the peanuts, mint, basil, 1 teaspoon salt, and ½ teaspoon pepper.
Toss and divide among individual plates.