Boil a big pot of water. when boiling add a tablespoon of salt. Add the broccoli florets and boil for 5 minutes. Take out the florets (keep the boiling water) and put them in a bowl with ice water to stop the cooking.
Put in the food processor the florets and all the other ingredients except for oil. Process shortly. Add the oil and process for a little longer, but make sure you keep some texture to the pesto.
cook the pasta in the water you used for cooking the broccoli. cook to a little shy of al dente, the pasta will finish cooking in the sauce. Keep one cup of pasta water before you drain the pasta.
Cut the roasted red pepper to thin ribbons.
Put 2 tablespoons of olive oil in a pan on medium high heat. Add the garlic and the pepper and saute for few minutes.
Add the pesto and 1/2 a cup of the pasta water and mix.
Add the cooked pasta, mix gently and let it cook for 2 minutes or so. If the sauce is too dry add a bit more water.
Remove from the heat and add the shredded basil and Parmesan cheese (as much as you like) and serve.