How to Make It
Heat 3 tablespoons of the olive oil in a large skillet. Add the garlic and cook over moderate heat until softened, about 1 minute. Add the eggplants and 2 tablespoons of the basil and season with salt and pepper. Cook over moderately high heat until browned, 6 to 8 minutes. Cover and cook over very low heat, stirring, until the eggplants are tender, about 3 minutes longer. Remove from the heat and let cool.
In a medium bowl, whisk the vinegar with the remaining 4 tablespoons of olive oil. Add the eggplants, tomatoes, cheese and the remaining basil, season with salt and pepper, toss and serve.