Boil the broccoli in hot salty water until tender. When ready, remove them from the water with a slotted spoon and keep aside. Use the broccoli water to cook your pasta in.
Mince the garlic and saute it in the olive oil in a frying pan large enough to combine the pasta and the sauce for a couple of minutes.
mash the broccoli with a potato masher or a wooden spoon and add to the garlic. Also adding a couple of tablespoons of the broccoli water. Add the half and half and stir. Let cook for about five minutes. Add 1/2 of the cheese and stir and turn off the sauce it should be thick.
Add enough water to your broccoli water and a teaspoon of salt to cook your pasta.
Add the pasta to the sauce and cook for 1 to 2 minutes. Garnish with rest of the cheese.