Put the beets, beet greens, onion, carrot, potato, and cabbage in a large pot. Add the vegetable stock and bring to a gentle boil over medium-high heat. Cover, reduce the heat to medium-low, and simmer until the vegetables are tender, about 20 to 30 minutes.
Pour in the tomato juice and enough beet kvas to give desired tartness. Stir in the dill and season to taste with salt and black p