How to Make It
In a saucepan, combine the water and sugar and stir over high heat until the sugar dissolves. Cook without stirring until a light amber caramel forms, about 2 minutes. Off the heat, add the vinegar. Cook the caramel over very low heat just until it softens, about 1 minute. Add the tomatoes, garlic and 1/4 cup of the oil and season with salt and white pepper. Simmer over moderately low heat until thickened, about 30 minutes.
Meanwhile, preheat the oven to 400°. Brush a large rimmed baking sheet with 2 tablespoons of the oil. Arrange the eggplant on the sheet and brush with the remaining 2 tablespoons of oil; season with salt and white pepper. Roast the eggplant for 10 minutes, then flip and roast for 10 minutes longer, until tender and lightly golden.
Arrange the eggplant slices in an 8-by-11-inch baking dish in slightly overlapping rows. Spoon two-thirds of the tomato sauce on top and bake for 20 minutes, until bubbling and browned around the edges. Sprinkle the cheese and mint on top and serve warm or at room temperature, passing the remaining sauce on the side.