Bring a large pot of salted water to a boil. Add the pasta and cook as the label directs. Reserve 1/2 cup cooking water, then drain.
Meanwhile, heat the olive oil in a large skillet over medium-high heat. Add the sausage and cook, breaking it up with a wooden spoon, until lightly browned, about 2 minutes. Add the onion and chard stems; cook, stirring occasionally, until slightly softened, about 3 minutes. Add the garlic and cook, stirring, until slightly softened, about 1 minute. Add the tomatoes and season with salt and pepper. Cook until the tomatoes start breaking apart, about 5 minutes.
Add the chard leaves to the skillet and cook, stirring occasionally, until they start wilting, about 3 minutes. Add the pasta, the reserved cooking water and 1/4 cup cheese; toss to coat. Season with salt and pepper. Top each serving with the remaining cheese.