Bring a large pot of salted water to a boil for cooking the pasta. Meanwhile, heat the oil in a large skillet over medium heat. Add the sausage; cook, breaking it up and stirring with a wooden spoon, until browned and cooked through, 7 to 10 minutes. With a slotted spoon, remove the sausage and set aside. If the pan seems a bit dry, add enough oil so there's about 3 tablespoons fat in the pan. Return the pan to medium heat, add the onion, and cook until just tender, 3 to 5 minutes, stirring and scraping up any browned bits with a wooden spoon. Add the chard, garlic, and pepper flakes, season with salt and pepper, and toss until the chard begins to sizzle, about 1 minute. Add the broth, cover, and simmer, stirring occasionally, until the chard is tender, 6 to 8 minutes. Meanwhile, cook the pasta until just tender. Reserve 1 cup of the pasta water and then drain the pasta, leaving drops of water clinging to it. Add the pasta and 1/4 cup of the cooking water to the skillet, using more of the water if needed. Return the sausage to the pan and add the balsamic vinegar. Heat gently for a few minutes to let the flavors meld. Add the pine nuts and half of the Parmigiano, and taste for salt and pepper. Add a bit more of the cooking water if the pan seems dry. Serve warm, passing the remaining Parmigiano at the table.