Line a baking sheet with parchment paper. Wrap each mozzarella slice with prosciutto until mostly covered. Put on the prepared baking sheet and cover with plastic wrap. Freeze until firm, about 30 minutes.
Preheat a grill to medium high. Grill the prosciutto-wrapped mozzarella until the prosciutto is crisp and the cheese just starts to melt, about 1 minute per side.
Arrange the prosciutto-wrapped mozzarella and sliced tomatoes on a platter, overlapping slightly. Top with the cherry tomatoes. Drizzle with the olive oil and vinegar and season with salt and pepper. Top with the basil.