Bring a large pot of water to a boil for the pasta.
Heat EVOO over medium-high heat and render pancetta or guanciale 2-3 minutes. Add onions, garlic and black pepper, and stir 7-8 minutes. Add tomato paste and stir a minute more. Deglaze the pan with vermouth or wine; add tomatoes and basil. Break up tomatoes, reduce heat and let simmer.
Salt boiling water and drop pasta. Cook 3 minutes then add broccoli and cook 5 minutes more. Reserve 1 cup starchy cooking water and drain pasta and broccoli. Combine pasta and broccoli with sauce and add starchy cooking water as necessary to combine. Add in a handful of cheese and serve.