In a small bowl, mix together the soy sauce, hoisin sauce, rice vinegar, and sesame oil. Set aside.
Heat a tablespoon of oil in a large skillet over medium heat. Add the minced garlic and ginger and cook for 1 minute until fragrant.
Add the ground chicken to the skillet and cook, breaking it up with a spoon, until browned and cooked through, about 5-7 minutes.
Add the chopped onion, bell pepper, carrot, and water chestnuts to the skillet. Cook for an additional 3-4 minutes, or until the vegetables are tender.
Pour the sauce mixture over the chicken and vegetables in the skillet. Stir well to coat everything evenly with the sauce.
Cook for another 1-2 minutes, allowing the sauce to thicken slightly.
Spoon the ground chicken mixture into lettuce leaves to create wraps. Serve with cooked rice on the side if desired.
Garnish with sliced green onions and sesame seeds, if desired.