Peel the eggplant and cut each crosswise into 1/4-inch-thick slices. Cover the bottom and sides of a large colander with a few eggplant slices and sprinkle generously with salt. Top with more layers of eggplant and salt until you run out of slices (you'll end up with five or six layers). Let the colander sit in the sink or over a large bowl for at least 30 minutes and up to 2 hours. The salt will draw out water and reduce the eggplant's ability to absorb oil.

Heat the 3 tablespoons oil in a 10-inch skillet over medium heat. Add the garlic and cook until fragrant and barely golden, 1 to 2 minutes. Add the tomatoes and 1/2 teaspoon salt. Raise the heat to medium high and cook, stirring occasionally, until the tomatoes begin to break down into a sauce, 20 to 25 minutes. If the sauce begins to dry up before the tomatoes break down, add warm water 1 tablespoon at a time. Lower the heat to medium and continue cooking, stirring occasionally, until you have a thick, chunky sauce, 5 to 10 minutes more. (Too much liquid in the sauce will make the finished dish watery.) Turn off the heat, remove the garlic, and stir in the basil leaves. Season to taste with more salt, if necessary, and set aside.

Dry the eggplant by lining a large plate with a paper towel and setting a few slices on it. Top with another paper towel and layer on a few more slices. Repeat until you run out of slices.

Attach a candy thermometer to the side of a 3- or 4-quart saucepan. Add the olive oil and heat over medium-high heat. When the oil reaches 375 degrees F, add as many eggplant slices as will fit comfortably in a single layer. Don't crowd the pan. If you don't have a candy thermometer, you can test the oil temperature by dipping a tip of one eggplant slice in the oil. If it immediately sizzles, the oil is ready.

Cook, turning once, until golden brown on both sides, about 2 minutes on the first side and 1 minute more on the second. Working quickly, pick up each slice with a slotted spoon and press the back of another large spoon against the slice to squeeze out as much oil as possible. Transfer to a plate lined with paper towels. Repeat until all the slices are fried, layering the fried eggplant between paper towels and adjusting the heat as necessary to maintain the frying temperature.

Position a rack in the center of the oven and heat the oven to 450 degrees F.

Layer about one-third of the eggplant slices so they overlap slightly on the bottom of a 10x8-inch (or similar size) baking dish. With the back of a spoon or an offset spatula, spread about one-third of the tomato sauce in a very thin layer over the eggplant. Evenly sprinkle about half of the mozzarella and 1/3 cup of the Parmigiano over the tomato sauce. Make another layer with one-third of the eggplant, one-third of the tomato sauce, the remaining mozzarella, and 1/3 cup Parmigiano. Make one last layer with the remaining eggplant, tomato sauce, and Parmigiano. Bake until the cheese has melted evenly and the top is bubbly, with browned edges, 20 to 25 minutes. Let rest for at least 15 minutes before serving.