Bring a large pot of water to a boil and add the penne. Cook at a rolling boil until the pasta is just tender. Drain, rinse under cold water, drain again, and reserve.
Cut the broccoli florets into fairly small pieces.
Bring a large saucepan of water to a boil and add the broccoli and simmer for 2 minutes. Drain, rinse under cold water, pat dry, and reserve.
Pour the oil into a large skillet, and heat over medium heat until it begins to ripple, about 1 minute. Add the garlic slices and cook, shaking the pan, until the garlic begins to brown around the edges, another minute.
Add the broccoli to the skillet, stir well, sprinkle with black pepper, and cook 2 minutes longer, shaking the skillet.
Add the butter and penne to the broccoli and cook, stirring often, until the penne is well mixed with the broccoli, oil, and garlic and the mixture is hot, about 3 to 4 minutes.
Place in a serving dish, sprinkle with the Parmesan cheese. Serve immediately.