1. Light the grill or heat the broiler. In a large glass or stainless-steel bowl, toss the zucchini with 1 tablespoon of the oil, 1/4 teaspoon salt, and 1/8 teaspoon pepper. Grill or broil the zucchini, turning once, until tender and golden, about 5 minutes per side. Put the zucchini back in the bowl.
2. Toss the zucchini with 1/2 tablespoon of the oil, 1/8 teaspoon salt, the vinegar, garlic, and parsley. Let cool.
3. Put the mozzarella slices on a serving plate, fanning them out to form a circle. Drizzle them with the remaining 1/2 tablespoon oil and sprinkle them with a pinch of pepper. Fold the zucchini slices in half and tuck them between the pieces of cheese.
Fresh Mozzarella: Soft mozzarella is shaped into balls and stored in water to keep it moist. It is available both salted and unsalted, but the latter is very bland indeed. We prefer the salted variety; salting brings out the mild, milky flavor of the cheese.