Heat the butter in a 4-quart saucepan over medium heat. Add the celery and onions and cook until tender, stirring occasionally.
Stir the broth, water, black pepper and potatoes in the saucepan and heat to a boil. Reduce the heat to low. Cover and cook for 15 minutes or until the potatoes are tender. Remove the saucepan from the heat.
Place half the broth mixture and 3/4 cup milk into a blender or food processor. Cover and blend until the mixture is smooth. Pour the mixture into a medium bowl. Repeat with the remaining broth mixture and milk (or you can skip these steps and use an immersion blender to blend the broth mixture and milk right in the pot). Return the broth mixture to the saucepan and cook until the mixture is hot, stirring occasionally. Season to taste. Sprinkle with additional sliced green onions before serving, if desired.