Start by caramelizing the onions. Split 2 onions in half, then cut each half into 1/4 or 1/2 inch slices. (You don't want super skinny slices for caramelizing.)
Heat a large, high-sided skillet over medium heat. Add 2 teaspoons olive oil. When the oil is hot, add the sliced onions and saute for 8-10 minutes. Reduce the heat to medium low. Add a dash of salt over the onions.
Continue to cook for another 35-40 minutes at least, checking on them every 10 minutes. If the onions start to stick to the pan and look dry, add another teaspoon of oil. In the last 10 minutes, add 1 or 2 teaspoons of water to deglaze the pan. Scrape the pan to work up all the browned bits. When the onions are as brown and sweet as you like them, remove the onions to a small bowl and set aside (keep warm)
Meanwhile, cook the pasta. Bring 2 quarts of water to a boil over high heat. Add the tablespoon of salt.
When it is boiling, add the pasta. Bring it back up to a boil, stirring occasionally for about 4 minutes less than the lowest suggested cooking time.
Meanwhile, prepare the broccoli by chopping off the stems and cutting the florets into similar sizes.
When the pasta has 3 minutes left, add the chopped broccoli and put the lid on. Set a timer for 3 minutes and don't lift the lid!
Drain the pasta and broccoli together in a colander. If your cheese sauce isn't ready yet, drizzle the pasta with a little olive oil and stir, so they don't stick.