Heat broiler. Cook the bulgur according to the package directions. Arrange the eggplant slices on 2 baking sheets. Brush both sides with a total of 3 tablespoons of the oil. Broil the eggplant, 1 sheet pan at a time, until brown, 2 to 3 minutes per side. Transfer to a large bowl. Stir in the tomatoes, vinegar, the remaining 2 tablespoons of oil, 1/2 teaspoon salt, and 1/4 teaspoon pepper. Fold in the basil. Transfer the bulgur to a large bowl and top with the eggplant and tomato mixture before serving.
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