Peel the eggplants and cut into one-inch dice. Sprinkle with salt and let stand in a colander for about an hour to drain off any bitter juices. Rinse and pat dry.

Pour olive oil to a depth of one-fourth inch in a large, wide, preferably nonstick sauté pan and warm over medium-high heat. In batches, cook the eggplant, turning often and adding oil as needed, until the eggplant is golden and cooked through, about eight to 10 minutes. Using a slotted spatula, transfer to paper towels to drain. Set aside. 

Pour off any oil remaining in the pan and place over medium heat. Add two tablespoons olive oil. When hot, add the celery and sauté briefly. The celery should still be a little crisp. Transfer to a plate.

Add olive oil to the pan as needed to measure two to three tablespoons and warm over medium heat. Add the onions and cook, stirring until tender but not brown, about eight minutes. Add the garlic and tomatoes or tomato purée and return the celery to the pan. Simmer for 10 minutes. Add the cooked eggplant, the vinegar, sugar, capers, olives, and pine nuts. Continue to simmer, uncovered, over low heat, for 20 minutes. Season with salt and pepper. Transfer to a serving dish and let stand for several hours, or preferably overnight. Adjust the seasonings. Serve at room temperature.