Rinse the chicken and pat it dry with paper towels. Pound it to an even thinness. Place the chicken in a large saucepan and add enough water to cover by 1/2 inch. Add 3 teaspoons of the salt and 1/2 teaspoon of the pepper and bring to a gentle simmer. Cook until no trace of pink remains, 8 to 10 minutes. Transfer the chicken 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 the chicken and pat it dry. Dice the chicken and add it to the apple mixture along with the peanuts, mint, basil, and the remaining salt and pepper. Toss and divide among individual plates.