How to Make It
In a large pot, melt the butter over moderately low heat. Add the onion; cook, stirring occasionally, until translucent, about 5 minutes.
Add the garlic, broccoli stems, potatoes, broth, water, salt, and pepper. Bring to a boil. Reduce the heat and simmer until the vegetables are almost tender, about 10 minutes.
In a food processor or blender, pulse the soup to a coarse puree. Return the soup to the pot and bring to a simmer. Add the broccoli florets and simmer until they are tender, about 5 minutes. Stir 1/4 cup of the grated Parmesan into the soup, and serve the soup topped with the remaining cheese.
If you'd prefer a completely smooth soup, add the broccoli florets to the pot after the potatoes have cooked for five minutes, and continue simmering until all of the vegetables are tender, about five minutes more. Puree the soup until smooth. This would also make a great first course for six people.