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 the grated Parmesan into the soup and serve.