Truth be told, I didn’t add any parmigiano reggiano to this recipe since mozzarella does such a great job here, and it achieves the desired effect. However, you can certainly add some Parmesan with the cheese layer during assembly if you’d like (but remember: only the real thing, freshly grated!). It also helps brown the dish if you sprinkle it on the top layer at the end, though the mozzarella also browns nicely.
Tomato Sauce (Garlic) *
3 firm medium eggplants , cut into 1/2″ rounds**
Extra virgin olive oil 
1 lb mozzarella, coarsely shredded or sliced
1 oz fresh basil, stems removed and chopped
Sea salt 
Prepare the tomato sauce and let it cool while you prepare the eggplant.
Preheat the oven to 350 degrees. Generously brush each side of the eggplant round with olive oil and place them on baking sheets. Take turns baking each sheet in the middle of the oven for 10-15 minutes or until the eggplant flesh is soft when poked with a fork. Be careful not to let the eggplant brown; you may have to flip some halfway through. Transfer to a large plate to cool. Continue until all the eggplant rounds are cooked and cooled.
Preheat the oven to 425 degrees and assemble the dish. Spread a layer of tomato sauce on the bottom of a large baking dish. Add a layer of eggplant without overlapping them. Sprinkle the eggplant with sea salt and fresh basil. Add a layer of mozzarella (and Parmesan, if you’d like). Repeat two more times or until all the eggplant has been used, ending with the sauce and finally the cheese. Bake for 25 minutes or until golden and bubbling. Let sit for 5 minutes and serve.
*I also added half of a chopped onion to the sauce with the garlic. It’s very important to use not only high-quality tomatoes (preferably San Marzano ) but fresh basil, which makes the sauce fragrant and sweet. A bitter or bland tomato sauce will ruin the dish.
**Discard any eggplant rounds with an abundant amount of seeds, which add bitterness. I keep the skin on the eggplant both for texture and health reasons (it contains an antioxidant  that prevents cell damage)