Cook beets in large pot of boiling water until very tender and knife pierces through to the center easily, about 50 minutes. Drain beets; cool slightly. Peel and coarsely chop.
Transfer beets, scallions, cucumbers, horseradish, and wine vinegar to food processor. Add some of the liquids to blend. Puree, but not too fine. Thin to desired consistency with beet juice and broth. Transfer to bowl, season, and chill several hours. Check seasoning before serving. Pour into individual bowls and garnish with spoonful of sour cream and sprinkling of dill. Can be made up to 3 days in advance, covered, chilled.