Bring 6 quarts of salted water to a boil in a large pot.
While the water is coming to a boil, make the sauce. Place all sauce ingredients in a high-speed blender and blend for about a minute or two, or until completely smooth, using a rubber spatula to scrape down the sides every 20 seconds or so. (If you don’t have a high-speed blender, see tip above)
Cook pasta in boiling water for about 5 minutes, then add the chopped broccoli. Boil until pasta is cooked, 3 to 5 minutes.
Drain pasta in a large colander and add immediately back to the pot. It should still be piping hot and wet with pasta water. Do not rinse and do not wait. This part is important because you need the wet, hot pasta to get the sauce creamy and awesome and clinging to the pasta.
Add the sauce to the pasta pot and use the rubber spatula to mix. Turn the heat on low and stir for about 2 minutes to get everything warmed through. Taste for salt and seasoning and serve.