Preheat an electric sandwich press or a stovetop grill pan.
Brush zucchini slices with oil. Place slices, a few at a time, on sandwich press; close lid. Grill 3 to 5 minutes, or until tender (if using an open grill pan, turn slices halfway through grilling). Continue grilling zucchini slices in batches until all are cooked. Transfer zucchini to a plate; season with salt and pepper.
Brush insides of bread slices with some of the Italian dressing. For each sandwich, place 2 zucchini slices on bottom slices of bread, folding to fit. Top with 3 slices prosciutto, then slices of provolone cheese. Place 2 zucchini slices over cheese, then top with 2 tomato slices. Drizzle with remaining dressing. Top with 3 basil leaves and dressed bread tops.
Grill sandwiches in sandwich press or on grill pan with a weight on top 3 to 5 minutes, until pressed and crusty. Halve sandwiches on the diagonal.