Slice eggplant crosswise into 1/3″ slices. Pat dry.
Whisk eggs and flour together in a shallow bowl.
Combine crumbs, seasonings in another shallow bowl.
Dredge eggplant slices in egg mix, then breadcrumbs. Lay on a rack to “dry” at room temperature 10-30 minutes.
Meanwhile, slice rolls open and place on baking sheet. Lay two slices cheese on half of each and put into 200ºF oven to melt cheese and warm buns.
Heat 2 tablespoons oil in a large skillet over medium heat and fry eggplant slices in a single layer about 3-5 minutes on each side until golden and tender.
As slices are cooked, drain on a rack and sprinkle with Parmesan while still hot. Cook remaining eggplant, adding a little more oil if necessary.
Once all slices are cooked, drain all but 1 teaspoon of oil from the skillet and CAREFULLY add sauce to skillet and heat to bubbling.
Remove rolls, top cheese with eggplant slices and sauce and fresh basil leaves.