Put eggplant in a colander set over a large bowl; sprinkle with 2 tsp. salt, tossing to evenly cover eggplant. Let stand 30 minutes.

Meanwhile, in a large Dutch oven or heavy pot with lid, heat 3 tablespoons oil over medium-high heat. Add onion, reduce heat to medium and cook, stirring occasionally, until onion softens, about 5 minutes. Add tomatoes, basil and 1/4 teaspoon salt. Reduce heat to medium-low; cook, covered until mixture is tender and aromatic, about 20 minutes. Remove from heat.

Rinse eggplant in colander under cold running water, then squeeze dry in small handfuls.

In a large skillet, heat 3/4 cup oil over medium-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 20 minutes per batch. Using a slotted spoon, transfer as cooked to paper towels to drain.

Transfer tomato mixture and juices to a bowl. Add 2 tablespoons oil to pot and heat over medium-high heat. Add olives, celery and capers; cook, stirring occasionally, 2 minutes. Add tomato mixture and eggplant; cook 1 minute more. Add nuts, raisins, vinegar and sugar; cook, stirring occasionally, 3 minutes more. Remove from heat; let cool completely. Serve caponata at room temperature, garnished with basil leaves, and with lightly toasted bread. You can also serve caponata mixed into pasta or over polenta.