Fill a large pasta pot about three-quarters full with water, add the salt and put on high heat. Cover. Once the water comes to a full boil, add in the broccoli. Give it a stir and let it come back to a full boil.
Add in the pasta and let it come back to a full boil. Turn heat down to medium/high heat. Cook uncovered.
Cook until it’s just under al dente because it will continue to cook as you assemble the mixture. Remember to reserve the cup of the pasta water. This is important because it helps make a sauce with the broccoli and EVOO/garlic mixture.
Drain the pasta. Keep the stove unit heat on because you’ll need to return the pot to the stovetop.
Add the drained pasta and broccoli back to the pan. Pour in the garlic and EVOO mixture. Make sure you get all the EVOO mixture by swirling a few pieces of pasta into the small saucepan and then pour it into the pasta pot.
Add in the grated cheese, black pepper and crushed red pepper. Put the pot back on the stovetop, add in half the reserved pasta waterand carefully mix it together. While gently mixing, break up the broccoli chunks with the side of the spoon.
Add in the rest of the pasta water if needed and stir until mixed through.