In a medium saucepan, heat the sugar and salt with the vinegar until dissolved. Then add the whole beets and enough water to cover them. Cook over medium-low heat for 40 minutes, or until tender.
Remove the cooked beets from the cooking liquid using a slotted spoon and set aside to cool, reserving the liquid. Put half the radish slices in some of the beet cooking juices. After half an hour, they will be pink.
Cut the cooled beets into fine slices. Arrange some in overlapping layers in the middle of a large, round plate. Arrange the radish slices and remaining beet slices in concentric circles around the plate.
Mix the dressing ingredients together and drizzle over the salad. Dot with bright green pistachio nibs and blood-red pomegranate seeds.