Whisk egg and 1 tablespoon water in small bowl to blend. Whisk in 2 tablespoons onions, 1/2 tablespoon soy sauce, and 1/2 teaspoon sesame oil. Heat 1 teaspoon vegetable oil in small nonstick skillet over medium-high heat 1 minute. Add egg mixture and tilt skillet to coat bottom. Cook until omelet is set, about 1 minute. Turn out omelet; cut into strips. Add broccolini, carrots, and remaining 1/3 cup water to same skillet. Cook uncovered over medium-high heat until vegetables are just tender and water evaporates, stirring often, about 3 minutes. Heat wok or large nonstick skillet over high heat 1 minute. Add remaining 2 teaspoons vegetable oil, ginger and garlic. Stir 20 seconds. Add broccolini, carrots, rice, and tofu. Toss to blend. Add vinegar, remaining 2 1/2 tablespoons soy sauce, and 2 teaspoons sesame oil. Toss until mixture is heated through, about 3 minutes. Add omelet strips and remaining 1/2 cup green onions; toss 1 minute.