Cut the eggplant into 5 mm / ¼ inch slices. Arrange the eggplant slices in a large colander, sprinkle with coarse sea salt, cover with a plate and put a weight on it, to allow the eggplants to sweat some of their water. Let them sit for 30 minutes, then rinse them and pat them dry with kitchen paper.
In the meantime, prepare the tomato sauce. Heat a sauce pan with the olive oil, add the onion and sauté until translucent. Add the tomato passata and a couple of basil leaves. Simmer on low heat for 20-25 minutes, stirring every now and then. Season to taste with sea salt and freshly-cracked black pepper and set it aside.
Heat the vegetable oil in a large frying pan oil until it reaches a temperature of 350°F / 175°C. Add a few eggplant slices at a time and fry until golden, then remove them and place them on kitchen paper to absorb the oil in excess.
Preheat the oven to 350°F / 175°C and arrange the baking tray onto the middle shelf.
Spoon a bit of the tomato sauce over the base of a baking dish. I use a 10.5" x 7"/ 26cm x17cm dish, a smaller dish will work well too.
Arrange a layer of eggplant slices, then cover with a few spoonfuls of tomatoes sauce, some grated parmesan cheese, and scatter a few basil leaves on top. Repeat with a layer of eggplant slices, then tomato sauce, parmesan cheese and basil leaves, until you have used up all the eggplant slices.
Finish with a layer of tomato sauce, sprinkle some parmesan cheese and optional breadcrumbs and mozzarella cheese on top. Bake in the oven for 30 minutes, then remove the parmigiana di melanzane from the oven, allow it to rest for 15-20 minutes and serve.