Bring a large pot of salted water to a boil. Meanwhile, heat 2 tablespoons olive oil in a large skillet over medium heat. Add the sausage and cook, breaking up the meat with a wooden spoon, until browned, about 3 minutes. Add the scallion whites and rosemary and cook until the scallions begin to wilt, about 2 minutes. Add the tomatoes with their juices and 1⁄2 cup of the boiling water. Let simmer while you cook the pasta.
Add the pasta to the pot of boiling water and cook as the label directs for al dente. Reserve 1/2 cup cooking water, then drain.
Add the pasta to the tomato sauce along with the spinach and half of the scallion greens. Toss to coat the pasta and wilt the spinach, about 1 minute, gradually adding the reserved pasta water as needed to loosen. Season with salt, drizzle with the remaining 1 tablespoon olive oil and remove from the heat.
Sprinkle the cheese over the pasta and toss well. Divide among plates or shallow bowls and top with the remaining scallion greens.