How to Make It
Heat the oil in a large heavy skillet over moderate heat. Add the onion and garlic and cook, stirring occasionally to keep the garlic from sticking, until softened, about 5 minutes.
Push the vegetables to one side of the skillet and add the tomato paste to the center. Cook, stirring, until a shade darker, 1 to 2 minutes. Add the red pepper flakes, then stir everything together and cook for 1 minute.
Add the eggplant, stirring to coat with the onions and oil. Stir in the vermouth, then reduce the heat to moderately low, cover and cook, stirring occasionally, until eggplant is tender, about 30 minutes. Stir in the olives and capers and season to taste with salt. Keep warm, covered, over very low heat (the sauce will be thick and chunky).
Meanwhile, cook the pasta in a large pot of well-salted boiling water until just al dente. Drain in a colander, reserving 1 cup of the pasta cooking water. Stir the pasta into the eggplant sauce, along with some of the reserved cooking water and the Pecorino. Season to taste with salt and pepper.