Remove stems and leaves from beets, but leave on skins. In a deep pot, cover beets with cold water and bring to a boil. Boil until beets are tender (approx 40 minutes).
Drain beets, but reserve two cups of liquid. Strain liquid and add to a large saucepan. Remove skin from beets. Grate beets through coarse grater. Add to beet liquid. Add beef broth, onion, salt, pepper and vinegar. Bring to a boil, and then cover and reduce heat to low. Simmer for 20 minutes and then remove from heat.
Chill in refrigerator for one hour, or until cold. Ladle into bowls and top each serving with diced cucumber and squeeze sour cream on top as desired.