Cook noodles according to package instructions; drain noodles and rinse with cold water. In a small bowl, combine soy sauce, sesame oil, and sugar and stir until sugar dissolves.
Heat a large skillet or wok over high until hot. Add 3 teaspoons vegetable oil and swirl to coat skillet. Add eggplant and stir until golden on both sides, 4 minutes total. Push eggplant to side of skillet and add 1 teaspoon oil. Add scallion whites and garlic and stir until fragrant, 30 seconds. Add noodles and soy mixture and toss to combine all the ingredients. Serve topped with scallion greens and basil.