Cut off the beet greens and reserve. Wash and scrub beetroots thoroughly, then peel them, and grate them into a bowl. Sprinkle the balsamic vinegar on top of the beets and let stand for 5-10 minutes.
Put the beet mixture into a large pot and cover with 6 qt cold water. Bring to a boil, add salt, then reduce heat and cook for 10 minutes.
Meanwhile, separate the beet stalks from the leaves. Chop the stalks finely and chiffonade the leaves or chop them into 1” pieces. Add the beet stalks to the pot after about 10 minutes of cooking. If using celery, dice it finely and add to the pot now as well. Cook for 5 minutes longer and add the beet leaves and herbs. Cook until the grated beets are soft but not falling apart, about 45 minutes to an hour.
Taste and adjust for salt and vinegar. If it tastes bland, you can add 1 tsp sugar. Chill.
Serve cold. To serve, put the chopped hard boiled egg into a soup bowl with some cucumber, radish, and scallions, then ladle the chilled soup on top. Garnish with parsley, dill, and sour cream, if using.