Trim the ends off the eggplant and cut it into 1/2-inch dice. Place the eggplant in a colander set over a plate or in the sink and sprinkle generously with salt. Let stand for 1 hour. Quickly rinse the eggplant pieces. Place the pieces in paper towels and squeeze until dry.
In a large deep skillet over medium heat, heat about 1/2-inch corn oil. To test the oil, carefully place a small piece of eggplant in it. If it sizzles and cooks rapidly, the temperature is hot enough. Add enough eggplant to make single layer. Cook, stirring occasionally, until crisp and browned. Remove the eggplant pieces with a slotted spoon. Drain well on paper towels. Repeat with the remaining eggplant. Set aside.
In a medium skillet over medium heat, cook the olive oil with the garlic and green onions for 30 seconds, or until fragrant. Add the swordfish cubes, salt, and pepper. Cook, stirring occasionally, until the swordfish is no longer pink, about 5 minutes. Add the vinegar and cook for 1 minute. Add the tomatoes and oregano. Bring to a simmer and cook for 10 minutes, or until slightly thickened.
Meanwhile, bring a large pot of cold water to a boil. Add salt and the cavatelli. Cook, stirring occasionally, until al dente, tender yet firm to the bite. Drain well.
In a large heated serving bowl, combine the cavatelli, sauce, and eggplant. Toss well. Stir in the cheese. Sprinkle with oregano or parsley and serve immediately.