Trim greens and stems from beets. Thoroughly wash and chop greens and stems. Peel and cut beets into 1/2-inch cubes.
Bring beets, chicken broth, leeks, celery root, carrots, and parsnips to a boil in a large pot. Add potatoes, reduce heat to medium-low, and simmer until vegetables are tender, 10 to 12 minutes. Stir beet greens and stems into soup and simmer until greens and stems are tender, about 5 minutes. Mix sugar, lemon juice, and salt into soup.
Stir sour cream and flour together in a bowl until smooth and thickened. Whisk 1 cup soup into sour cream mixture until smooth; pour sour cream mixture into soup and stir. Simmer soup until thickened slightly, about 1 minute. Add dill and remove pot from heat. Ladle soup into serving bowls and arrange 4 egg quarters in each bowl.
You can make this soup up to 3 days ahead. Cool the soup uncovered, then cover and chill. Reheat, stirring occasionally, over medium-low heat. Wait to add the lemon juice until just before serving, otherwise it will turn your veggies a grayish-khaki color as they sit in the fridge.
Vegetable broth can be used in place of chicken broth, if desired.
Per Serving: 310 calories;9 g fat;
44 g carbohydrates;
15 g protein;
222 mg cholesterol;
903 mg sodium.