Cut off the stem of lettuce, remove outer pieces of lettuce and throw away. Pull apart leaves and rinse with cold water. Set aside.
Mix together all ingredients for Thai sauce in small bowl. Set aside.
Heat the oil in large sauté pan over medium heat. Add the onion and ginger and cook, stirring frequently, until soft, 4-5 minutes. Add the ground chicken and turn the heat up to high. Cook, breaking up the meat with a wooden spoon, until partially cooked through, about 5 minutes. Add the garlic and cook 1 minute more.
Add the Thai sauce to the chicken and cook for a few more minutes. Take off heat. Stir in the scallions, cilantro, mint, peanuts, and shredded carrots. Taste and adjust seasoning if necessary. Spoon into lettuce cups and serve with a lime wedge