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1
Peel eggplant and cut lengthwise into 1/8-inch-thick slices.
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2
Sprinkle sea salt over slices and arrange in 2 even stacks in a shallow dish.
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3
Put a flat dish on top of eggplant and weight with 1-lb can 2 hours.
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4
Rinse salt from eggplant under running water and pat slices dry between double layers of paper towel.
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5
Preheat oven to 350F.
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6
Heat oil in an ovenproof 12-inch heavy skillet over moderately high heat.
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7
Saute eggplant in batches of 3 or 4 slices until golden, about 2 minutes per side.
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8
Drain as sauteed on brown paper or paper towels.
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9
Pour off oil from skillet and wipe, leaving a light coating.
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10
Cover bottom of skillet with half of eggplant, overlapping slightly, and make another layer with remaining eggplant.
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11
Whisk together eggs and cheese in a bowl and stir together tomato paste and water in another bowl.
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12
Heat skillet with eggplant over moderate heat until hot and pour in egg mixture, lifting edges of eggplant so eggs coat bottom of skillet.
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13
Cook omelet, uncovered, over low heat, until eggs begin to set around edge, 5 to 7 minutes.
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14
Slowly pour tomato mixture in a spiral pattern over surface of omelet, then bake in upper third of oven until eggs are set and pale golden, about 20 minutes.
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15
Loosen side and bottom of omelet and slide onto a plate.
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16
Serve at room temperature with a few grindings of black pepper.
