1. Wash the eggplants, cut off both ends, and slice them very thinly lengthwise.
2. Salt each slice and let them rest covered, to release the vegetable's juices.
3. Fry each slice in a large skillet and let them brown on each side. Season the tomato sauce with salt and pepper.
4. In the serving dish, layer the eggplant slices with a little sauce, fresh basil, and Parmesan cheese.
5. Repeat the layers, arranging the slices in a different direction each time to make serving easier.
6. Refrigerate, as this dish is served cold.