Peel the eggplant and cut into lengthwise slices no more than 3/8-inch thick. Purge the eggplant slices for 30 minutes or more. When purging is done, rinse the eggplant and pat each slice thoroughly dry with paper towels.
Preheat oven to 400 degrees.
In a large frying pan, pour enough oil into it to come 1 1/2-inches up the sides, and turn the heat up to high. Dredge the eggplant in the plate of flour and slip as many slices of eggplant into the pan as will fit loosely without crowding. Cook to a golden brown color on one side, then turn them and do the other side. Do not turn them more than once. Drain on a cooling rack and/or paper towels. Repeat the procedure until all the eggplant is done.
Put the tomatoes and olive oil in another skillet, turn the heat onto medium-high, add salt, stir, and cook the tomato down until it is reduced by half.
Smear the bottom of a 11x7-inch oven-to-table baking dish with oil. Put in enough fried eggplant to line the bottom of the dish in a single layer, spread some of the cooked tomato over them, cover with a layer of mozzarella, sprinkle liberally with grated Parmesan cheese, distribute a few pieces of basil over it, and repeat layers, ending with a layer of eggplant on top. Sprinkle with grated Parmesan cheese, and place the dish in the upper third of the oven.
Cook for 35 minutes; checking halfway through to discard any excess liquid that may be thrown off.
Allow dish to settle for several minutes before serving.
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