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1
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.
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2
Add garlic and saute, stirring, 1 minute.
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3
Add tomato puree, 1/2 teaspoon salt, and 1/4 teaspoon pepper and simmer, uncovered, stirring occasionally, until slightly thickened, about 30 minutes.
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4
Gently pound chicken breasts to 1/3-inch thickness between 2 sheets of plastic wrap using a flat meat pounder or a rolling pin.
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5
Season with salt and pepper.
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6
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.
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7
Dredge chicken slices in flour, shaking off excess, then dip in egg, letting excess drip off, and dredge in bread crumbs until evenly coated.
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8
Transfer chicken to sheets of wax paper, arranging slices in 1 layer.
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9
Preheat oven to 400F.
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10
Heat vegetable oil in a deep 12-inch heavy skillet over moderately high heat until hot but not smoking, then Fry chicken 3 pieces at a time, turning over once, until golden brown and just cooked through, about 6 minutes per batch.
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11
Transfer with tongs to paper towels to drain.
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12
Arrange bottom halves of bread loaves on a large baking sheet and tops on another large baking sheet, all with cut sides up.
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13
Spread 1/4 cup tomato sauce on each top and bottom.
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14
Divide chicken between bottom halves.
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15
Top each (open-faced) sandwich with 1/4 cup tomato sauce and one fourth of mozzarella.
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16
Bake open-faced sandwiches in lower third of oven until cheese melts, about 3 minutes.
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17
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).
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18
Put tops on bottoms to make sandwiches, then slice into serving pieces.
