Melt the butter in a large saucepan over medium-low heat. Add the scallions and cook for 1 minute.
Add the potatoes, wine, cream, broth, 1 ¼ teaspoons salt, and ¼ teaspoon pepper and bring to a boil.
Reduce heat and simmer until the potatoes are tender, about 15 minutes.
Ladle the soup into individual bowls and sprinkle with ¼ teaspoon pepper.