Bring large pot of water to boil. Blanch broccoli florets 2-3 minutes or until stems are fork tender. Remove with slotted spoon. Pour in pasta to same water from broccoli. Cook 7-10 minutes or until al dente.
While pasta is cooking, sauté onions and garlic in olive oil over medium heat for about 5 minutes, or until browned and softened.
Spoon cooked onions and garlic into the bowl of a food processor with cooked broccoli, cheese, a little pasta water {or heavy cream} and salt and pepper. Pulse until it starts coming together, adding in more pasta water or cream to get it to the consistency you'd like. I liked mine chunky, so I didn't add too much. Taste and adjust seasonings.
Drain pasta and toss with broccoli pesto. Grate more cheese overtop and serve warm.