Bring a large pot of salted water to a boil. When water boils, add pasta and broccoli at the same time and cook according to pasta instructions for al dente. When pasta is almost done cooking, reserve about 1 cup of the pasta water and set aside. Drain pasta and broccoli.

Return

 the pot to the stove and set heat to high; 

add 

1 tbsp olive oil, when hot, 

add

 garlic. 

Cook 

until golden, 

reduce

flame to low and 

add

 pasta back to the pot. Mix well, add remaining olive oil, grated cheese, salt and pepper to taste mixing well and smashing any large pieces of broccoli to break up.

Add 

1/2 cup of reserved pasta water and 

mix well

 adding more if needed.

Serve

 in pasta bowls with additional grated cheese on the side.