As soon as it's done, drain the pasta in a colander (having one of those colander inserts that fits inside your pot is a big help). Don't rinse it! Just shake off some of the excess water.
You can even cook the broccoli or whatever greens you're using at the same time in the same pot in which you're cooking the pasta. This is blanching -- cooking just long enough to make things tender. The trick is figuring the timing, and that is based on the density of the vegetable. Soft, leafy greens can be cooked in a minute or two; tough, woody stems can take as long as the noodles do.
In fact, with broccoli, add the diced stems at the same time you start the pasta and then add the florets when there is only a minute or two left.
The only thing left to make is the sauce, which for this particular pasta is essentially just olive oil and flavorings. This can be as simple as minced garlic by itself, or you can layer in other ingredients.
One obvious way to go is adding some kind of pork product, mainly because adding some kind of pork product will make almost anything taste better. Italian sausage is good -- either sweet or hot. Squeeze it out of its casing and break it into bite-sized chunks as you add it to the pan. Flatten it slightly with your fingers to increase the surface area so it will brown better (browned pork products are even better than plain pork products).
But remember that this sauce is really just seasoning. Anything besides raw sausage -- salted anchovies, olives, capers, whatever you like -- needs only to be heated through.
Watch the garlic
The one thing you want to be careful with is cooking the garlic. Don't let it scorch. If you see it beginning to go from golden to brown, immediately pull the pan from the heat and add a spoonful of the pasta water to stop the cooking.
The final fillip in the sauce-making part of the recipe is adding a ladleful of the pasta cooking water back into the saucepan. This accomplishes two things. Because there is so much starch in the water, it slightly thickens the sauce. More important, it marries the flavors of the sauce to the noodles.
To finish the dish, you just need a little cheese. Parmesan is a natural, of course, but I prefer pecorino Romano, which has a sharp, slightly funky sheep's milk edge to it. Grate the cheese if you want, but I like it better sliced off in thin shards with a vegetable peeler. That way you get a nice, distinct bite, rather than little bits mixed in with everything else.
But you should try it both ways and make your own decision. That's the beauty of learning to make a dish, rather than just following some recipe.
VARIATIONS ON A THEME: The recipe calls for sausage, but prosciutto will work too. Broccoli is a no-brainer, but try the Chinese kind for a change of pace. Finally, you can go meatless with capers and olives, unless you want anchovies too.