Bring a pot of salted water to a boil. Add the pasta and cook as the label directs. Meanwhile, heat the olive oil in a large skillet over medium heat. Add the sausage and cook, breaking it up with a wooden spoon, until it begins to brown, about 3 minutes. Add the spring onions and cook until softened, about 3 more minutes. Add the garlic, mushrooms, 1/2 teaspoon salt, and pepper to taste; cook, stirring occasionally, until the mushrooms start to brown, about 3 minutes. Add the cherry tomatoes and cook until they soften slightly, about 3 more minutes.
Reserve 1 cup of the pasta cooking water, then drain the pasta. Add the broth to the skillet and cook, stirring, 1 minute. Stir in the pasta and cheese, adding some of the reserved cooking water to loosen, if necessary. Season with salt and pepper. Divide among bowls, drizzle with olive oil and top with more cheese and spring onions.