['2 lb small Italian eggplants (about 4)', '1 tablespoon kosher salt or 2 teaspoons fine sea salt', '1 1/2 cups extra-virgin olive oil', '1 cup chopped onion (1 medium)', '4 medium celery ribs, cut crosswise into very thin', '1/3 cup large green Sicilian olives (1 3/4 oz), pitted and coarsely chopped', '1 3/4 oz Italian capers packed in salt (1/3 cup), rinsed well', '2 tablespoons sugar, or to taste', '1/3 cup white-wine vinegar', '1 (14- to 15-oz) can whole tomatoes in juice', 'drained and chopped (1 cup)']
Peel eggplants, leaving some strips of peel, then cut into 1-inch cubes and spread on half of a kitchen towel. Sprinkle eggplant with salt, then cover with other half of towel and weight with a baking sheet topped with 2 or 3 large cans for 30 minutes.
Meanwhile, heat 1/2 cup oil in a 10-inch heavy skillet over moderately high heat until hot but not smoking, then cook onion, stirring, until pale golden, 6 to 8 minutes. Add celery and cook, stirring, until onion and celery are deep golden brown, about 10 minutes. Add olives, capers, and 2 tablespoons sugar and cook, stirring, 2 minutes, then stir in vinegar and tomatoes.
Reduce heat and simmer, covered, stirring occasionally, 20 minutes. If sauce is very acidic, add 1 to 2 teaspoons sugar (to taste). Transfer to a bowl and keep warm, covered.
Rinse eggplant in a colander under running water, then squeeze dry in small handfuls.
Heat remaining cup oil in cleaned skillet over moderately high heat until hot but not smoking, then fry eggplant in 2 batches, turning occasionally with tongs, until tender and browned on all sides, 8 to 10 minutes per batch. Transfer as cooked to paper towels to drain, then transfer to a large shallow serving dish in an even layer. Spoon sauce on top, spreading evenly, and let stand, covered with a kitchen towel, at room temperature, at least 8 hours (for flavors to develop). Stir before serving.
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