Rinse the broccoli and pat it dry. Cut off the thick stalks (and discard them) so that you have florets with 2-3 inches of the stems attached. Slice any thicker stems lengthwise in half. Slice your garlic into thin rounds.
Heat olive oil in a large pasta pot on medium high. Add garlic and cook for two minutes, stirring so the garlic doesn't burn.
Lower the heat to medium. Noting the time, add broccoli, pepper flakes and ¼ teaspoon salt and stir. As soon as the oil evaporates, add ½ cup water (see notes) and stir.
Cover the pot and let the broccoli cook until it is soft, stirring occasionally. It should take about 25 minutes from the time you added the broccoli to the pot.
When the broccoli is soft, add 5 cups of water to the pot. Add in ⅓ teaspoon salt and give it a quick stir. Cover the pan and increase the heat to high. When the water is boiling, add the pasta and cook, uncovered, stirring frequently, until it is al dente. (Check for doneness three minutes before the al dente time listed on the package.)
When the pasta is cooked, do not drain it. Turn off the heat and add in ⅓ cup Parmesan, stirring to combine.
Serve with extra Parmesan, a drizzle of oil and red pepper flakes. Store leftovers in the refrigerator for up to five days.
To reduce the amount of olive oil needed, I incorporated the step of adding ½ cup of water to the pot with the broccoli to allow it to cook until soft without burning. If you prefer to use more olive oil instead, just eliminate the ½ cup of water and add enough extra oil to cook the broccoli until it's soft.