Melt butter in a large saucepan over medium heat. Stir in onion and sauté until tender. Stir in potatoes and radish greens, coating them with butter. Pour in chicken broth; bring to a boil. Reduce heat and simmer for 30 minutes.

Allow soup mixture to cool slightly, then transfer to a blender. Blend until smooth.

Return soup to the saucepan; stir in heavy cream until warmed through. Serve with radish slices.