Begin preparing the salads at least one day ahead to allow the flavors to blend.
Serves 8. Can be doubled.
Stir vinegar, shallots and 6 teaspoons sugar in small bowl to blend. Let marinade stand while preparing vegetables.
Cook beets in large pot of boiling salted water until tender, about 30 minutes; drain. Peel beets. Cut into wedges. Transfer to medium bowl. Toss with 1/2 cup of marinade to coat.
Place cucumbers in large bowl. Sprinkle 4 teaspoons sugar over. Toss with remaining marinade. Season salads to taste with salt and pepper. Cover separately and refrigerate 1 day, stirring occasionally. (Can be prepared 2 days ahead. Keep refrigerated.)
Drain salads separately; return to bowls. Mix half of chives into each salad. Arrange salads on platter and serve.