Remove stems and leaves from beets, but leave on skins. Cover beets with cold water and bring to a boil in a deep pot. 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 the most coarse blade of a grater. Add to beet liquid. Add beef broth, onion, vinegar, salt, and pepper. Bring to a boil, and then cover and reduce heat to low. Simmer for 20 minutes, then remove from the heat.

Chill in the refrigerator for 1 hour, or until cold, before serving. Ladle into bowls and top each serving with cucumber and a big spoonful of sour cream.