Trim the brocolli. Divide into small florets and cut the stalks.
Bring a large saucepan of water to a boil. Add about 1 1/2 tbsp of cooking salt and then slide in the brocolli. Stir well and cook for 5 minutes after the water has come back to a boil. Retrieve the brocolli from the water with a slotted spoon and lay them on kitchen paper. Pat dry and set aside.
Bring the brocolli water back to the boil and add the pasta. Cook in the usual way until al dente.
While cooking the pasta, chop the garlic, chili and anchovy fillets together and saute them in half the oil for 2 minutes, using a large frying pan. Mix in the brocolli and saute for a few minutes, turning constantly.
When the pasta is done, drain and turn it into the frying pan. Stir-fry for 1 minute, then taste and check the seasoning.
Before you serve the pasta, pour over the rest of the olive oil and mix in the freshly grated parmesan.
