Peel the eggplant and cut it into thin rounds, not more than 1 cm (1/8 inch) thick. Sprinkle the slices with salt and lay them in a colander placed inside a bowl or on top of a plate. Place a small plate on top of the eggplant and then weigh the plate down with something not too heavy like a small can.

Let the eggplants rest for a good hour or so, then pat them dry and lay them out on a cutting board. Then cut the ham slices and cheese slices into rounds slightly smaller than the eggplant slices.

Now make "sandwiches" by placing one slice of ham and one of cheese in between two slices of eggplant.

Proceeding as if you were making a cutlet, dip these sandwiches in flour, then in the beaten egg—which you will have seasoned with salt and pepper and, if you like, a bit of grated Parmesan cheese—then finally in the breadcrumbs. At each step, make sure to cover the sides as well as the tops and bottoms of the sandwiches. If there are any bald spots, you can repeat the process.

In a large frying pan, fry the eggplant slices in olive oil, enough to come about 1 cm (1/8 inch) up the sides of skillet. The oil shouldn't be too hot, just hot enough to bubble gently around the sides of the sandwiches. Take your time with it. Otherwise, the outsides will brown before the eggplant has time to fully cook.

Lay the fried slices out on a baking rack or on a platter lined with paper towels. When you've finished frying all of them, serve them right away.