Bring a large stockpot of salted water to a boil. Blanch the broccoli – cook more than al dente stage, but not to a mush. Remove and shock in ice water to stop cooking. Keep water at a simmer to use the same water to cook cavatelli.
Heat a large sauté pan with enough capacity to contain all of the sauce, broccoli and pasta over medium heat. Add the olive oil and garlic, and cook over low heat, just to flavor the oil with the garlic. The garlic should not burn.
Meanwhile, cook the cavatelli per instructions to al dente. Drain into a bowl, and add broccoli, some cheese and the garlic oil. Toss to combine and serve with plenty of grated Parmesan on the side.