In a large skillet, heat the 3 tablespoons of olive oil. Add the onion and garlic and cook over moderate heat until tender, about 5 minutes. Using your hands, crush the whole tomatoes into the skillet. Bring to a simmer and cook over moderately low heat, stirring occasionally, until the sauce is very thick, about 25 minutes. Transfer the tomato sauce to a food processor and puree until smooth. Season with salt and pepper.

Meanwhile, in a very large skillet, heat 1/4 inch of olive oil. Season the eggplant slices with salt and pepper. Working in several batches, cook the eggplant over moderately high heat, turning once, until golden on both sides, about 8 minutes per batch; add more olive oil to the skillet between batches. Drain the eggplant slices on paper towels.

Preheat the oven to 400 degrees. Spread 1 cup of the tomato sauce in a 9-by-13-inch glass or ceramic baking dish. Arrange one-third of the fried eggplant slices in the baking dish and sprinkle all over with 1 tablespoon of the chopped basil. Top with one-third of the torn mozzarella and sprinkle with 1/3 cup of the grated Parmigiano-Reggiano. Repeat this layering twice. Sprinkle the bread crumbs all over the top of the eggplant Parmesan. Bake in the upper third of the oven for about 45 minutes, until the top of the eggplant Parmesan is golden and the tomato sauce is bubbling. Let stand for 15 minutes before serving.

The assembled eggplant Parmesan can be covered with plastic wrap and refrigerated for up to 1 day. Bring the eggplant Parmesan to room temperature before baking.