Bring a medium pot of salted water to a boil. Meanwhile, heat the oil in a large, heavy skillet over medium-high heat until shimmering hot. Add the sausage and cook, stirring occasionally, until browned, about 3 minutes. Add the mushrooms, scallions, rosemary, red pepper flakes, 3/4 tsp. salt, and 1/2 tsp. pepper and cook, stirring often, until the mushrooms soften and start to brown, about 3 minutes. Add the tomatoes and chicken broth, bring to a boil, and then cover and reduce to a gentle simmer. Cook until the sausage is heated through and the flavors are melded, about 5 minutes.
Meanwhile, cook the pasta according to package timing until it’s just al dente. Drain well and add to the sauce along with half of the Parmigiano. Cook over medium heat, tossing for 1 minute. Serve sprinkled with the remaining Parmigiano and some black pepper.