Toss the eggplant cubes with kosher salt. Place the salted eggplant in a colander set over a bowl. Let it sit for about 30 minutes to draw out excess moisture. Rinse the eggplant well and pat it dry with paper towels.

Heat 2 tablespoons of extra virgin olive oil in a large skillet over medium heat. Add the finely chopped celery and cook, stirring often, until softened (about 4 minutes). Add the finely chopped onion and minced garlic; cook and stir until the onion is soft and lightly golden (about 5 minutes). Transfer the mixture to a bowl using a slotted spoon and set aside.

In the same skillet, heat the remaining 2 tablespoons of extra virgin olive oil. Add the eggplant and cook, stirring constantly, until lightly browned (about 5 to 7 minutes).

Stir in the cooked celery, onion, and garlic mixture. Add the coarsely chopped tomatoes, kalamata olives, capers, tomato paste, and dried oregano. Bring the mixture to a boil, then reduce the heat to low. Simmer uncovered until the caponata is thickened (about 15 minutes).

Season the caponata with red wine vinegar, white sugar, salt, and ground black pepper. Transfer the caponata to a serving bowl. Garnish with minced fresh parsley. Drizzle a little extra extra virgin olive oil on top before serving if desired.