Preheat the broiler. Bring a large pot of salted water to a boil. Add the pasta and cook as the label directs for al dente. Reserve 1/2 cup cooking water, then drain.
Meanwhile, heat the olive oil in a large pot over medium-high heat. Add the sausage and cook, breaking it up into pieces, until browned, 3 to 5 minutes. Add the onion and chard stems; cook, stirring, until softened, 3 to 5 minutes. Add the garlic and cook 30 seconds. Add the tomatoes, oregano, 1/4 teaspoon salt and a few grinds of pepper.
Cook, stirring, until thickened, about 4 minutes. Add the chard leaves and cook, stirring, until softened, about 3 minutes. Add the pasta and 1/4 cup reserved cooking water to the sauce and stir to coat, adding more cooking water as needed to loosen; season with salt and pepper. Transfer to a 3-quart baking dish. Dollop spoonfuls of the ricotta on top, then sprinkle with the mozzarella. Broil until golden brown and bubbling, 3 to 5 minutes. Let rest 5 minutes before serving. Sprinkle with parmesan.