Prepare the Sauce: In a small bowl, whisk together the soy sauce, fish sauce, brown sugar, lime juice, sesame oil, and cornstarch. Set the sauce aside.
Cook the Filling: Heat oil in a large non-stick skillet over medium-high heat. Add the light parts of the green onions, garlic, and ginger, sautéing for about 30 seconds. Then, add the ground chicken and cook, breaking it up with a spoon until it's nearly cooked through (no need to drain).
Add Vegetables: Stir in the diced red bell pepper and cook for another 2 minutes until it softens slightly. Then, add the carrots to the skillet.
Add the Sauce: Give the sauce a quick whisk again and pour it over the chicken mixture in the skillet. Toss everything together and cook for about 1 minute, or until the sauce thickens and the chicken is fully cooked. Taste and season with a little salt if necessary.
Finish the Dish: Stir in the cilantro, green onion greens, and peanuts.
Serve: Spoon the chicken mixture into whole lettuce leaves, and top with hot sauce, extra peanuts, and peanut sauce if desired.