Melt butter in a large saucepan over medium heat. Add onion; cook 5 minutes or until tender, stirring frequently. Add radishes and potatoes, tossing to coat with butter. Stir in broth and milk; bring to a boil. Cover, reduce heat, and simmer 15 minutes or until potatoes are tender. Cool 10 minutes.
Place half of radish mixture in a blender; process until smooth. Pour pureed soup into a large bowl. Repeat procedure with remaining radish mixture. Cover and chill at least 2 hours.
Add sour cream, salt, pepper, and nutmeg, stirring with a whisk. Cover and chill at least 4 hours or up to overnight. Sprinkle with chives; garnish with radish slices, if desired.