Preparation Heat 2 Tbsp. oil in a large skillet over medium-high. Cook sausage, breaking up with a wooden spoon, until cooked through, browned, and crisp in places, 5–8 minutes. Transfer to a small bowl with a slotted spoon. Reduce heat to medium; add onion and olives to skillet. Cook, stirring often, until onion is translucent, about 5 minutes. Add tomatoes, increase heat to medium-high, and cook, mashing tomatoes lightly and stirring often, until juices are slightly thickened, about 3 minutes. Meanwhile, cook pasta in a large pot of lightly salted boiling water, stirring occasionally, until very al dente, about 6 minutes. Drain, reserving 1 cup pasta cooking liquid. Add pasta and 3/4 cup pasta cooking liquid to sauce and cook, tossing, until pasta is al dente and sauce coats noodles. Mix in peperoncini, cooked sausage, and another 1/4 cup pasta cooking liquid. Then, tossing constantly, gradually add all but 1/2 cup cheese, followed by butter. Once incorporated, remove from heat and mix in basil. Divide pasta among bowls. Drizzle with oil; top with remaining cheese.