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 fork tender, about 40 minutes.
Drain beets, but reserve two cups of the liquid. Strain the liquid and add to a large saucepan. Remove skin from beets. Grate beets through coarsest blade of 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, then remove from the heat.
Chill in refrigerator for one hour, or until cold before serving. Ladle into bowls and top each serving with cucumber and a big spoonful of sour cream.
