Fill a large pot with water, add 1 tablespoon of salt and bring to a boil. When boiling, add the chopped broccoli and cook for 3-4 minutes.
Drain the broccoli and divide into 3/4 and 1/4.
To the same pot add the pasta and cook according to package instructions.
Meanwhile make the sauce: in a large pan heat olive oil over medium heat. add crushed garlic and cook for 2 minutes. Add 3/4 of the blanched broccoli and cook for 2 minutes. Add 1/2 cup of pasta cooking water and blend into smooth puree.
Put the pan back to medium heat, add the remaining broccoli chunks, salt, black pepper and drained pasta. Add 1/4 cup of pasta cooking water and toss to combined.
Add grated cheese and toss until melted and creamy. Add more pasta cooking water as needed.