How to Make It
Light a grill or heat the broiler. Brush the eggplants with olive oil and season with salt and pepper. Grill over moderately high heat, turning occasionally, until charred and softened, about 15 minutes. Transfer to a cutting board and coarsely chop. Brush the cut sides of the rolls with olive oil and grill on both sides until lightly toasted, about 1 minute.
Meanwhile, in a large bowl, whisk the 1/2 cup of olive oil with the vinegar and garlic. Season with salt and pepper. Add the eggplant, celery, celery leaves, parsley, chives and basil and toss to coat. Mound the salad on the rolls and top with the Parmesan. Close the sandwiches, cut them in half and serve.