1. In a 3- to 4-quart enameled or stainless-steel saucepan, bring the grated beets and cold water to a boil over high heat.
2. Reduce the heat to moderate and cook uncovered for 10 minutes. Then reduce the heat to low, stir in 2 tablespoons of the vinegar, 2 teaspoons of the salt and 1 teaspoon of the sugar, and simmer partially covered for 30 minutes.
3. Dram the beets in a fine sieve set over a large bowl. Set the beets and the cooking liquid aside separately to cool to room temperature.
4. Peel the shrimp. Devein them by making a shallow incision down their backs with a small, sharp knife and lifting out the black or white intestinal vein with the point of the knife.
5. Wash the shrimp under cold running water. Then bring 1 quart of water to a boil in a small pan, drop in the shrimp and cook briskly, uncovered, for about 3 minutes, or until they turn pink and are firm to the touch.
6. Drain and coarsely chop the shrimp. Set aside to cool. When the beet cooking liquid is completely cooled, beat in the sour cream with a wire whisk.
7. Then stir in the beets, shrimp, cucumbers,scallions, radishes, 2 tablespoons of the dill, the lemon juice, the remaining tablespoon of vinegar, 3 teaspoons of salt and ½ teaspoon of sugar, and a pinch of white pepper.
8. Taste for seasoning, cover the bowl tightly with plastic wrap, and refrigerate for at least 2 hours, or until the chlodnik is thoroughly chilled.
9. To serve, ladle the soup into a large chilled tureen or individual soup plates. Sprinkle the remaining dill on top and, if you like, garnish the chlodnikwith thin slices of lemon and shopped hard-cooked eggs.
