Heat the olive oil in a 12-inch sauté pan, add the cubed eggplant, and fry until golden brown.
Set aside on a plate lined with paper towels.
Preheat a stovetop grill.
Place the sliced eggplant on a clean surface and sprinkle generously with salt.
Marinate for 5 minutes, then pat them dry with a clean kitchen towel.
Brush each with 1/2 teaspoon of olive oil.
Place the eggplant on to the grill and cook for 3 minutes on each side, or until you get nice grill marks. Set aside and let them cool.
Preheat the oven to 375°F.
In a food processor, combine the diced eggplant, ricotta, basil, Parmigiano, pepper, and nutmeg. Blend until smooth.
Lay the grilled eggplant slices flat and spread about 1 tablespoon of the mixture evenly on each.
Add 1 cube of mozzarella cheese and roll the eggplant from bottom to top, ensuring the mixture stays in the middle of the roll. In a large baking dish, add 2 tablespoons of tomato sauce and place the rolls side by side in the dish.
Top the rolls with the rest of the tomato sauce, sprinkle with Parmigiano, and bake for 25 minutes.