Put vinegar in a medium saucepan over high heat, bring to a boil and reduce until thick and syrupy. Stir in the honey and coarsely ground black pepper and transfer to a bowl. Can be made 1 day in advance and refrigerated and brought to room temperature before using.
Heat the grill to high.
Brush both sides of the eggplant with oil and season with salt and pepper. Place the eggplant on the grill and grill until golden brown and slightly charred, about 4 minutes. Turn over and continue grilling until just cooked through, 4 to 5 minutes longer.
Transfer 8 slices of the eggplant to a flat surface and top each slice with a slice of cheese and 2 basil leaves and season with salt and pepper. Repeat, making 4 stacks, each with 3 slices of eggplant and 2 slices of cheese. Carefully return the stacks to the grill and grill for 2 minutes, or until the cheese is slightly melted. Remove the stacks to a large dinner plate and drizzle with some of the balsamic-black pepper glaze. Garnish with some of the chopped basil.