Bring a large pot of water to a boil over high heat and add a generous amount of kosher salt. Stir in the trennette, return the water to a boil, and cook the pasta, stirring occasionally to prevent it from sticking together, until it's al dente. (Since cooking times vary, refer to the package instructions for the recommended time and taste the pasta for doneness frequently while it cooks.)

While the water is coming to a boil and the pasta is cooking, heat the olive oil, garlic, and a pinch of kosher salt in a medium skillet over medium-high heat until the garlic is soft and fragrant, about 1½ minutes, stirring constantly so the garlic doesn't brown. Turn the heat up to high, add the tomatoes and eggplant and smash the eggplant with a wooden spoon. Cook them just until the ingredients are warmed through, about 3 minutes. Turn off the heat and season with kosher salt and freshly ground black pepper, keeping in mind the olives and ricotta salata will add salt to the dish.

Drain the pasta and transfer it quickly, while it's still dripping with water, to the skillet with the tomatoes and eggplant. Place the skillet over high heat and toss the pasta with the sauce for a minute or two, until the pasta is coated and the sauce is warm.

Spoon the pasta out of the skillet and pile it into high mounds on four plates, dividing it evenly. Sprinkle it with the olives and oregano and using a microplane, grate a generous amount of ricotta salata over each serving.