Bring a large pot of salted water to a boil. Add broccoli and return to a boil. Cook until tender, about 12 minutes. Drain well and transfer to a bowl. Add salt, pepper and olive oil. Mash roughly with a fork.
Remove casings from sausages and crumble meat into a large skillet. (Or cut into slices and brown sausages in casings.) Add 1/4 cup water and cook over medium-high heat, stirring, until water has evaporated, fat is rendered and sausage is no longer pink anywhere, 10 to 12 minutes.
In a pot of boiling salted water, prepare pasta according to package directions. Set aside 1/2 cup pasta water; drain pasta well and return to pot. Add broccoli, sausage, pimiento, Parmesan and 1/2 cup reserved pasta water. Toss to mix well. Serve with extra Parmesan.