Heat a large skillet over medium-high heat. Add chicken, garlic, ginger, coleslaw mix, basil, sugar, and pepper. Pour in hoisin sauce, soy sauce, and vinegar. Bring to a simmer, and cook and stir until the chicken is hot, and the liquid has been absorbed, about 10 minutes.
To serve, line a platter with lettuce leaves, and pour the chicken mixture into the center of the platter. Each diner assembles their own wraps by placing some of the chicken mixture onto a lettuce leaf, adding a little peanut sauce, then rolling into a cylinder.