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Place eggplant slices on a sheet tray and bake for 15 minutes, until light golden brown.
Form sandwiches out of the eggplant slices and cheese, placing one slice of cheese between two eggplant slices. Dredge each sandwich in the flour and dip in the egg. Heat the olive oil in a large skillet until almost smoking, and fry the sandwiches until golden brown. Remove to a plate lined with paper towels.
Arrange sandwiches in a ceramic pie pan and distribute the basil leaves evenly over the sandwiches. Season the tomatoes with salt and pepper to taste, and spoon them over the sandwiches. Sprinkle each sandwich with Parmigiano-Reggiano and bake for 17 to 20 minutes. Serve warm.