Put the eggplant in a colander in the sink, salt it generously and let sit for at least 30 minutes to release moisture, then press down on it with paper towels. Heat 2-3 tablespoons of olive oil in a large nonstick skillet over medium-high heat. Add the eggplant in a single layer (you may have do this in batches) and cook, tossing occasionally, until golden brown and fork tender, about 8 to 10 minutes and transfer to a bowl. Repeat with the remaining olive oil and eggplant. Return all the cooked eggplant to the pan and add the tomatoes, garlic, red pepper flakes, and some salt and pepper to taste. Cook, tossing (scraping up any tasty browned bits from the bottom of the pan) until the tomatoes are soft, about 2 to 3 minutes. Add the oregano and vinegar and give it a quick stir. Meanwhile, cook the pasta in boiling salted water according to the package directions or until al dente. Reserve ¼ cup of the cooking water. Drain the pasta and return it to the pot. Add the tomato and eggplant mixture to the pasta along with the reserved cooking water. Toss to combine. Sprinkle with any cheese you have on hand.