Remove stems and leaves from beetroot, but leave on skins. In a deep pot, cover beetroot with cold water and bring to a boil. Boil until fork tender, about 40 minutes.
Drain beetroot but reserve two cups of the liquid. Strain the liquid and add to a large saucepan. Remove skin from beetroot. Grate beetroot through the coarsest blade of a grater. Add to beetroot liquid.
Add beef stock, 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. Ladle into bowls and top each serving with cucumber and a big spoonful of sour cream.