Cover beets with water in a pot and bring to a boil; cook until tender about 15 minutes. Remove from heat and stir in chicken stock. Allow beets to cool.

Mix cucumber, egg, dill, and green onions in a bowl. Whisk buttermilk and sour cream until smooth in another bowl. Stir egg mixture and buttermilk mixture into cooled beets. Cover and refrigerate at least 2 hours before serving.