Preheat oven to 450 degrees. Trim the ends of the eggplant and slice crosswise into 1-inch thick rounds; cut each round into halves or quarters, to have roughly equal pieces, no bigger than 2-inches on a side. Put the chunks on a baking sheet, lined with parchment or a silicone baking sheet, and sprinkle over a tablespoon of oil and 1/4 teaspoon salt. Toss to distribute the oil and spread the pieces apart so they’ll brown quickly.

Put the tomatoes on another parchment-lined sheet, sprinkle over 1 tablespoon oil and a pinch of salt, roll them around and spread them out. Put both sheets in the oven and roast until both the eggplant and the tomatoes are soft, shriveled and nicely caramelized on the edges, 30 minutes or more. Turn the eggplant chunks while roasting a couple of times, roll the tomatoes over, and shift the sheets around in the oven for even heating.

Let the vegetables cool on the sheets completely, then transfer them to a large mixing bowl. Toss gently with the remaining olive oil and salt, ground pepper, vinegar, and basil. Taste and adjust the seasonings. Arrange the salad on a serving platter or portion on salad plates and sprinkle the shredded cheese on top.