The first time I made this, many years ago, I cooked the garlic & oil and the onions & broccoli first, then made the pasta. Now I have the timing down so I can do all three at once. Place a large covered pot of cold water on the stove and bring to a boil. Remove cover, add pasta and cook according to directions on the package. Remember to stir after the first 30 seconds, and then every few minutes, so the pasta does not stick to the pan or itself. If using whole broccoli stems, trim off the bottom half to two-thirds, leaving the tender crown. Trim off the florets with stems attached. Chop or slice the remaining stems. Peel onions, cut in half, then slice to create "crescent moon" shaped pieces. Separate garlic cloves, peel, then chop to medium fine pieces. It's okay if some pieces are larger than others, but none should be huge. Pour about six ounces of olive oil into a small sauce pan. Add the chopped garlic. Cook over medium-high, stirring very carefully once or twice. When the garlic just starts to change color, turn off the heat and move the pan off the burner. The garlic will continue to cook to a light golden brown as the oil cools. It is important to watch the oil and garlic closely. If you let it cook too long, the garlic will burn. Place onions into a large, deep frying pan with one or two tablespoons of oil. Saute over high heat until they start to caramelize. Add the broccoli, toss, add one ounce of water, then cover the pan and continue cooking. Remove the cover after about two minutes. The broccoli should have turned a bright green. Continue cooking, tossing occasionally, until the broccoli is tender, but slightly crisp. Turn off the heat. Replace the cover if holding for the pasta to be done. Drain the pasta. Return to pot or place in a large bowl. Add the onions & broccoli and toss together. Add the oil and garlic, 1/4 cup or so of dried basil and black pepper to taste, then toss until everything is coated with the oil. Place a dish of grated cheese on the table, along with some cayenne pepper, for those who would like to sprinkle either or both over the pasta. Serve hot with some homemade garlic bread and a good chianti. Variation: For a change, my mother would sometimes add breadcrumbs to the dish. If you want to try this, add one to two cups of your favorite breadcrumbs after tossing with the oil, then toss again. You will probably have to add some extra olive oil to the dish as the crumbs can otherwise make the dish a bit dry.