To make the sauce:

Pour the whole and stewed tomatoes into a bowl and break them up with a potato masher or with your hands.

In a large nonreactive saucepan, warm the oil and garlic over medium heat. Use a wooden spoon to press down on the garlic to release its aroma and swirl the pan to infuse the oil. Don’t let the garlic brown or it will become bitter. Carefully pour in the tomatoes, taking care to avoid spatters, and stir to coat with the oil. Add the salt, raise the heat to medium-high and bring the tomatoes to a simmer. When the juices start bubbling, reduce the heat to medium and let the tomatoes cook, uncovered, stirring from time to time, for 30 to 35 minutes, or until the sauce has thickened and the oil has separated from the tomatoes. Remove from the heat and stir in the basil. Cover and set aside.

Cut the eggplant lengthwise into ½-inch-thick slices. Sprinkle them on both sides with a little salt and layer them in a colander. Set a plate on top of the eggplant and weight it down with a heavy object. Let the eggplant drain for 30 to 60 minutes. Wipe the slices dry with paper towels.

Heat ¼ cup vegetable or olive oil in a large skillet or sauté pan placed over medium-high heat. When the oil is hot (375°F), add enough eggplant slices to fill the skillet without crowding it. Fry the eggplants until they are golden brown, taking care to turn them once so they brown on both sides, about 5 minutes total. Reduce the heat to medium if the eggplant seems to be browning too quickly. Use a large slotted spatula to remove the cooked eggplant to a platter lined with paper towels or a large brown paper shopping bag placed nearby. As the eggplant slices cool after draining, transfer them to a platter. Continue to fry the eggplant slices, adding more oil as necessary, until you have cooked them all.

To assemble and bake the parmigiana:

Heat the oven to 350°F. Have the sauce, eggplant, cheeses, and basil nearby. Layer about ½ cup of sauce into a large rectangular baking dish. Arrange one layer of eggplant slices over the sauce. Top with some slices of fresh and smoked mozzarella and sprinkle with some Parmigiano cheese and basil. Add 2 more layers of sauce, eggplant, mozzarella and Parmigiano cheeses, and basil. Finish with a layer of eggplant, topped with a final layer of sauce and a generous sprinkling of Parmigiano cheese.

Bake the eggplant for 30 to 40 minutes, until it is bubbling and the top is nicely browned. Remove from the oven and let sit for about 20 minutes to firm up a bit. Serve the parmigiana warm or at room temperature.

Cook’s Note: Fresh mozzarella contains a lot of moisture, which will be released when you bake the parmigiana. If you like a parmigiana that is less juicy, use the drier part-skim mozzarella available in most supermarkets in place of the fresh mozzarella.

Do Ahead: The eggplant can be fried several hours or up to a day in advance and stored in layers in a tightly lidded container in the refrigerator.