To cook the eggplant, place a large skillet over medium-high heat. Add 2 teaspoons of the cooking oil. When oil is shimmering, add ginger and cook until fragrant, turning once, about 30 seconds. Add eggplant spears, arranging them with cut sides down and peel up or to the side. Cook undisturbed until lightly browned, 30 seconds to 1 minute. Turn and cook another interior side until browned, about 1 minute. Toss well and add grape tomatoes. Cook, turning now and then, until eggplants are tender, softened, and nicely browned and tomatoes are shiny and starting to wilt, 1 to 2 minutes more.