In a small bowl, combine the coconut aminos, tomato paste, vinegar, sesame oil, garlic, and ginger. Whisk well to combine.
Place a large skillet over medium-high heat. When hot, add the avocado oil and swirl to coat the bottom. Add the ground chicken, mushrooms, and onion.
Season with salt and pepper. Break up the chicken with a wooden spoon and cook, stirring occasionally, until the chicken is almost cooked through, 4 to 5 minutes. Add the water chestnuts and continue to cook, stirring occasionally, until the chicken is cooked through, 3 to 4 minutes.
Decrease the heat to medium low. Add the sauce to the skillet and bring to a slow simmer. Continue to cook, stirring occasionally, until the sauce is mostly absorbed, 5 to 7 minutes. Add salt and pepper to taste.
Spoon the chicken-mushroom mixture into the lettuce leaves, layering two leaves if they are small, and serve with your desired toppings.
Store leftover chicken-mushroom filling in an airtight container in the refrigerator for up to 4 days.