Step 1

In a large pot, melt the butter over moderately low heat. Add the onion; cook, stirring occasionally, until translucent, about 5 minutes.

Step 2

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.

Step 3

In a food processor or blender, pulse the soup to a coarse puree.

Step 4

Return the soup to the pot and bring to a simmer. Add the broccoli florets and simmer until they are tender, about 5 minutes.

Step 5

Stir 1/4 cup of the grated Parmesan into the soup, and serve the soup topped with the remaining cheese.