Sprinkle both sides of the eggplant slices with salt, and set them aside on paper towels for about 30 minutes.
Meanwhile, prepare the sauce: Heat the olive oil and butter in a 2-quart saucepan over medium heat. When it is hot, add the onions and 1/4 teaspoon of the salt and sauté, stirring frequently, until the onions are nicely browned and caramelized, about 6 minutes. Add the garlic and cook until fragrant, about 30 seconds. Add the tomatoes and simmer until they are tender and the sauce is slightly thickened, about 12 minutes. Add another 1/4 teaspoon of the salt, the black pepper, the cayenne, and the chopped basil. Remove the sauce from the heat and set it aside to cool briefly. Then transfer the mixture to a blender, season it with 1/2 teaspoon of the salt, and puree (see Note). Set the sauce aside.
Quickly rinse the eggplant slices under cool run- ning water, and then pat them dry with paper towels.
Position an oven rack about 6 inches from the broiler unit, and preheat the broiler.
Add enough oil to a large straight-sided sauté pan to reach 1/2 inch up the sides, and heat it over medium-high heat until it reaches a temperature of 375°F. Line one or two baking sheets with paper towels, and set them aside.
While the oil is heating, place the flour and beaten eggs in separate shallow pans that are large enough to hold the eggplant slices. Combine the breadcrumbs and the remaining 1 teaspoon salt in a third pan; whisk to blend.
Dip an eggplant slice in the flour, coating both sides and shaking off the excess. Dip the floured slice into the eggs, coating both sides. Then finally dredge the slice in the breadcrumbs, again coating both sides. Set the breaded eggplant aside on a tray. Repeat with the remaining eggplant slices.
In batches, place the breaded eggplant slices in the hot oil, taking care not to overcrowd the pan, and cook until they are golden brown and crisp, about 5 minutes per side. Transfer the eggplant to the prepared baking sheet to drain. Continue in this manner until all of the eggplant is cooked.
Place the eggplant slices on a clean baking sheet. Top each slice with 1 to 2 tablespoons of the tomato sauce and some of the sliced mozzarella, dividing the mozzarella among the eggplant slices. Sprinkle 1 tablespoon of the Parmesan over each. Transfer to the broiler and cook until the cheese is melted and bubbly, 1 to 2 minutes.
Carefully layer the eggplant slices on top of one another on six serving plates. Top with more tomato sauce, and garnish with chopped basil and any remaining Parmesan. Serve immediately.
Yield: 6 servings
Note: Please use caution when blending hot liquids: blend only small amounts at a time, with the blender tightly covered and a kitchen towel held over the top.