In a large saucepan or Dutch oven over high heat, combine the beets, broth, and water. Bring to a boil. Reduce the heat to low, cover, and simmer for 30 minutes, or until the beets are very tender.
Finely chop the cooled beets and add to the cooking liquid. Cover and refrigerate until cold.
Add the lemon juice and vinegar to the soup. Place half of the soup in a blender or food processor and blend or process until smooth. Add to the soup with the cucumber.
Ladle the soup into bowls. In a cup, combine the sour cream and dill. Dollop onto each serving.