Mix chicken with 1 tbsp. fish sauce, the sugar, and 2 tbsp. ginger in a small bowl; set aside. In another small bowl, stir hoisin sauce with lime juice, peanut butter, and remaining 1 tsp. fish sauce until smooth; set sauce aside.
Heat a large frying pan (not nonstick) over medium-high heat. Swirl in oil. Add garlic and remaining 1 tbsp. ginger and cook until fragrant. Add chicken and cook, stirring and breaking up chunks, until no longer pink, 1 1/2 to 2 minutes. Add carrot, water chestnuts, green onions, lemongrass, and chiles and cook until chicken is cooked through, about 1 minute more. Add cucumber and peanuts and stir to combine. Add salt to taste. Transfer to a serving bowl.
To eat, spread some hoisin sauce on a lettuce cup. Top with chicken mixture, fold up, and eat with your hands.