To make tomato sauce: Heat olive oil in a 4 to 5-quart heavy saucepan over moderately-high heat until hot but not smoking, then saute onion, stirring occasionally, until golden, about 5 minutes. Add garlic and saute, stirring, 1 minute. Add tomato puree, 1/2 teaspoon salt, and 1/4 teaspoon pepper and simmer, uncovered, stirring occasionally, until slightly thickened, about 30 minutes.
To make the eggplant: Toss eggplant slices with 1 teaspoon salt in a colander set over a bowl, then let stand 30 minutes.
Stir together flour, remaining 1/2 teaspoon salt, and remaining 1/2 teaspoon pepper in a shallow bowl, then stir together bread crumbs and Parmigiano-Reggiano in another shallow bowl.
Dredge eggplant slices in flour, shaking off excess, then dip in egg, letting excess drip off, and dredge in bread crumbs until evenly coated. Transfer eggplant to sheets of waxed paper, arranging slices in 1 layer.
Preheat oven to 400 degrees F.
Heat vegetable oil in a deep 12-inch heavy skillet over moderately-high heat until hot but not smoking, then fry eggplant 4 slices at a time, turning over once, until golden brown, 2 to 3 minutes per batch. Transfer with tongs to paper towels to drain.
Arrange bottom halves of bread loaves on a large baking sheet and tops on another large baking sheet, all with cut sides up. Spread 1/4 cup tomato sauce on each top and bottom. Divide eggplant between bottom halves, overlapping slices slightly. Top each (open-faced) sandwich with 1/4 cup tomato sauce and one fourth of mozzarella.
Bake open-faced sandwiches in lower third of oven until cheese melts, about 3 minutes. When cheese begins to melt, put tops of loaves in upper third of oven and bake until edges are golden, 3 to 4 minutes (watch tops closely; they burn easily). Put tops on bottoms to make sandwiches, then slice into serving pieces.