Toss the eggplant with the kosher salt and spread out onto a microwave-safe plate. Cook in the microwave on High until the eggplant is dry and has shriveled to 1/3 of its original size, about 15 minutes. Transfer to a paper towel-lined plate and set aside.
Whisk the tomato juice cocktail, vinegar, red wine, brown sugar, parsley, and anchovies together in a large bowl. Stir in the tomatoes, raisins, black olives, green olives, zucchini, yellow squash, celery, red bell pepper, and carrot; set aside.
Heat 1 tablespoon of the olive oil in a large skillet over medium-high heat. Stir in the eggplant and cook, stirring occasionally, until the edges have browned, 4 to 8 minutes. Add the remaining 2 tablespoons of olive oil along with the celery and onion. Cook and stir until the onion has softened, about 4 minutes more. Stir in the vegetable mixture and bring to a simmer. Reduce heat to medium-low and simmer uncovered until the liquid thickens and begins to coat the vegetables, 4 to 7 minutes more. Scrape into a bowl and stir in the pine nuts. Refrigerate to room temperature. Sprinkle with the Parmesan cheese and serve.
