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1. Preheat oven to 375u00b0. Heat 1/4 cup olive oil and garlic in a medium saucepan over medium heat until garlic is fragrant, about 1 minute. Add tomatoes, season to taste with salt and pepper, and simmer, stirring, until sauce thickens, about 30 minutes.
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2. Meanwhile, place flour in a shallow dish. Beat eggs together in another shallow dish. Mix bread crumbs with a generous pinch of salt and pepper in a third shallow dish. Set dishes aside.
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3. Peel and trim eggplant and slice lengthwise into 1/2'' pieces. Dredge each slice first in the flour, then in the egg, then in the seasoned bread crumbs.
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4. Heat remaining 1/8 cup oil in a large cast-iron skillet over medium-high heat until oil is hot but not smoking. Add breaded eggplant slices to the hot oil (working in batches, if needed) and cook until golden on both sides and dark brown on the edges, 2-3 minutes per side.
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5. Spread a thin layer of tomato sauce in the bottom of a large shallow ovenproof dish. Arrange eggplant in a single layer on top of tomato sauce. Spoon remaining sauce over eggplant. Scatter basil on top of sauce and sprinkle with parmigiano-reggiano, then provolone. Bake until sauce is bubbling and cheese is melted, about 20 minutes.
