Place the beets in a large pot of boiling salted water and cook uncovered until the beets are tender, 30 to 40 minutes. Remove the beets to a bowl with a slotted spoon and set aside to cool. Strain the cooking liquid through a fine sieve and also set aside to cool.

In a large bowl, whisk together 1 1/2 cups of the beet cooking liquid, the chicken stock, sour cream, yogurt, sugar, lemon juice, vinegar, 1 tablespoon salt, and the pepper. Peel the cooled beets with a small paring knife or rub the skins off with your hands. Cut the beets in small to medium dice. Add the beets, cucumber, scallions, and dill to the soup. Cover with plastic wrap and chill for at least 4 hours or overnight. Season, to taste, and serve cold with a dollop of sour cream and an extra sprig of fresh dill.