Salt the eggplant on each side and let it drain in a colander for about 30 minutes to get the bitterness out. Place something heavy, like a frying pan, on top to get the moisture out.
In a medium bowl, place the flour, 1 teaspoon salt and 1 teaspoon pepper. In a second medium bowl, add the eggs, Parmigiano-Reggiano, half of the parsley and some salt and pepper, and whisk together. In a third medium bowl, add the breadcrumbs and the remaining parsley and toss together. Coat the eggplant slices in the flour, then in the egg mixture and then in the breadcrumbs.
Heat the vegetable oil in a skillet and fry the eggplant slices until they're crisp and golden brown. Don't crowd the pan. Let them drain on a paper towel to get excess oil off.
Top 1 slice of eggplant with 1 slice of tomato, and then season with salt, pepper and garlic powder. Add 3 basil leaves, top with extra-virgin olive oil and balsamic vinegar and a slice of mozzarella. Repeat the process with the remaining ingredients.
In a large bowl, toss the arugula with some salt, pepper and extra-virgin olive oil. Add to the top of the stacks and add more Parmigiano-Reggiano on top. Season with salt and pepper.