Adapted from Marcella Hazan. Essentials of Classic Italian Cooking.
Cut the broccoli tops into small florets. Peel the stems and cut them into roughly 1-inch pieces.
Heat the olive oil and butter in a large pot large enough for all the liquid over medium-low heat, then add the onion and season with salt and black pepper—cooking until translucent and just beginning to brown, ~8 minutes. Then add the garlic and cook for another minute.
Add the potatoes, broccoli stem, broth, and water. Season generously with salt and pepper. Bring to a boil, then reduce the heat and simmer until the potatoes are almost tender (potatoes should offer just a slight resistance when pierced), ~10 minutes.
Transfer the soup to a blender and blend to a smooth consistency—or, use a stick blender to blend the soup in the pot.
Return the soup to the pot and simmer. Add the broccoli florets and simmer until they are cooked, ~4 minutes. Then stir the Parmesan into the soup.