Stir chicken, water chestnuts, mushrooms, yellow onion, 1/3 cup green onion, 1 tablespoon soy sauce, ginger, and 2 teaspoons brown sugar together in a bowl. Pat mixture down, cover the bowl with plastic wrap, and refrigerate until ready to use.
Whisk chicken stock, rice wine vinegar, garlic, ketchup, 1 tablespoon soy sauce, sesame oil, 2 teaspoons brown sugar, red pepper flakes, and dry mustard together in a bowl until glaze is well-mixed.
Heat vegetable oil in a heavy nonstick skillet over high heat. Cook and stir chicken mixture in hot oil until chicken no longer looks raw, about 2 minutes. Pour 1/2 of the glaze mixture into chicken mixture; cook and stir over high heat until chicken begins to caramelize and brown, 10 to 15 minutes.
Reduce heat to medium-low, pour remaining glaze into chicken mixture, and cook until heated through and slightly reduced, about 3 minutes more. Stir in cilantro, basil, and 1 1/2 tablespoons green onion. Transfer chicken mixture to a bowl and serve with lettuce leaves to wrap.