Heat a gas grill to high, covered. Mince the garlic, sprinkle the salt over it, and scrape repeatedly with the flat side of a chef's knife until the garlic is mashed to a paste. Combine the garlic paste with the olive oil in a small bowl and add the thyme and a generous amount of pepper.

Trim the tops off each eggplant. Score the eggplant skin lengthwise with the tines of a fork. Cut each eggplant in half lengthwise and, using the sharp tip of a knife, score the cut sides in a crosshatch pattern, going halfway through the flesh, abut ½ inch deep. Gently knock out any loose seeds from the tomatoes. Put the eggplant and tomatoes in a glass baking dish large enough to hold them. Toss with the garlic oil and sprinkle them with pepper.

With the lid down, grill the eggplant on both sides until tender and deeply colored but not burnt, about 10 minutes. Grill the tomatoes, cut sides down, just until they're nicely warmed and slightly charred, 2 to 5 minutes. Baste the vegetables while grilling with the remaining seasoned oil. Mix the eggplant and tomatoes on a serving platter and scatter the basil and cheese over them. Drizzle more oil on top or pass a bottle of extra-virgin olive oil at the table.