Heat a large wok or sauté pan over medium-high heat. Add 1 tablespoon vegetable oil and sesame oil. Once hot, add the carrots and onions, and cook until crisp-tender, about 2 minutes. Add the water chestnuts, ¼ cup green onion, garlic, and ginger, and cook until fragrant, about 1 minute. Transfer to a medium bowl, and set aside.

Return the wok to the stovetop, and heat 1 tablespoon of vegetable oil over high heat. Add the ground chicken, break it into smaller chunks, and stir-fry until the meat is no longer pink, about 3 to 4 minutes—season with salt and pepper.

Stir in the hoisin sauce, soy sauce, and rice vinegar.

Add the stir-fried vegetables and mix evenly with chicken—season with salt and pepper to taste.

Evenly divide the chicken filling among the lettuce cups. Garnish with peanuts (if using) and remaining green onions. Top with additional hoisin sauce if desired.