In a small saucepan, boil the carrots in lightly salted water until very tender. Drain and rinse in cold water. Pat dry and set aside. Place the horseradish in a food processor with the metal blade and grind until coarsely grated. With the machine still on, using the feed tube, add the vinegar, salt and half the carrot juice. Process until the horseradish is finely grated, but not pureed. Carefully remove processor lid, averting your face when removing the lid to avoid the strong fumes, and scrape down the sides of the bowl.
With the machine still running, add the carrots alternately with the remaining carrot juice. Again, carefully remove the contents with the liquid to a bowl. Mix very well, cover tightly and refrigerate for at least 3 hours before serving. Before serving, stir again, adding more salt, vinegar and optional sugar, if desired. Yields about 11/4 cups to serve with gefilte fish or brisket.