Step 1Trim the root ends of the green garlic and the very tips of the green leaves if they are dried out. Cut the green garlic crosswise in thin pieces. Slice the potatoes in half lengthwise and then into about half-inch pieces. Place them in a bowl of water to prevent coloring.
Step 2In a large saucepan, combine the butter and onion and cook over medium heat, stirring roughly until the butter melts and the onions turn soft and creamy, about 2 to 3 minutes. Add the garlic and the green garlic, reduce the heat to low, cover, and cook until the mixture is fragrant, about 5 minutes.
Step 3Add the potatoes and turn them in the garlic mixture. Add the broth and salt, increase the heat to medium and bring to a simmer. Loosely cover and cook at a quick simmer until the potatoes are soft enough to be smashed with a fork, about 20 minutes.
Step 4Coarsely puree the potatoes and garlic. This is most easily done with an immersion blender but can also be done in a food processor or blender if you're careful to pulse quickly. The mixture should be chunky, not a smooth puree.
Step 5Add one-half teaspoon sherry vinegar and a generous grinding of black pepper. Taste and add more salt, pepper or vinegar if necessary. Return to the pan and simmer another 5 minutes. This makes about 7 cups of soup.
Step 6Stir briskly just before serving. Ladle into warm serving bowls, drizzle with a thread of olive oil and sprinkle over 1 to 2 tablespoons grated pecorino Romano.