Preheat oven to 400 degrees F. Lightly brush bottom of 2 sheet pans with olive oil. Arrange tomatoes cut sides up on one half of one pan and sausage on remaining side. Spread chard stems in one layer on one half of remaining sheet pan and place onions on other half. Season all with salt and pepper to taste. Roast vegetables and sausage in upper and lower thirds of oven, stirring onions and chard occasionally, until golden and tender, about 15 minutes for chard and onion and 30 minutes for tomatoes and sausage. Transfer vegetables, as cooked, to a large bowl. When cool enough to handle, quarter sausage lengthwise and cut crosswise into 1/2-inch pieces; add to bowl.
Meanwhile, cook pasta in a 4-quart pot of boiling salted water, stirring occasionally, until al dente, about 8 minutes. Reserve 1/2 cup pasta cooking water and drain pasta. Add pasta and chard leaves to bowl with cooking water and gently toss to combine. Transfer pasta mixture to a shallow 2-1/2-quart casserole dish and sprinkle top with cheeses. Bake pasta in middle of oven, uncovered, until golden on top, about 20 minutes.