Toss eggplant with 1/2 tsp. kosher salt and place in a colander set over a bowl. Let sit about 30 minutes. Rinse and pat dry.
Heat 2 Tbsp. of olive oil in a large skillet over medium heat. Add celery. Cook, stirring often until softened, about 4 minutes.
Add onion and garlic. Cook and stir until onion is soft and lightly golden, about 5 minutes. Transfer mixture to a bowl using a slotted spoon.
Heat remaining 2 Tbsp. of olive oil in the skillet. Add eggplant. Cook, stirring constantly until lightly browned, 5–7 minutes.
Stir in celery mixture, tomatoes, olives, capers, tomato paste, and oregano. Bring to a boil, reduce heat to low, and simmer uncovered until caponata is thickened, about 15 minutes.
Season caponata with red wine vinegar, granulated sugar, remaining salt, and pepper.
Transfer to a serving bowl and garnish with parsley. Once cooled to room temperature, fold in Fresh Mozzarella and serve.
Pair this savory caponata with an amber Marsala Rubino from the beautiful vineyards of Sicilia.