Soak the dried mushrooms in 1 cup hot water for 15 minutes. Drain, reserving the liquid, and squeeze out the excess liquid. Strain all the soaking liquid through a fine-mesh sieve. Reserve the liquid (there should be 1 cup) and the mushrooms separately.
In a tall narrow stockpot, heat the oil over medium heat. Stir in the fresh mushrooms and cook, stirring occasionally, for 4 minutes. Stir in the onion and cook, stirring occasionally, for 8 minutes.
Add the beets, carrots, parsnip, celery root, 8 cups water, and the mushroom soaking liquid. bring to a boil. Lower the heat and simmer for 5 minutes. Stir in the potatoes, cabbage, garlic, and, if using, the beet greens. Dissolve the tomato paste in 1/2 cup of the liquid and stir back into soup. Return to a boil. Lower the heat and simmer for 5 minutes. Stir in the reconstituted mushrooms and simmer for 5 minutes, or until all the vegetables are tender.
Remove from the heat. Stir in the dill, sugar, vinegar, salt and pepper.
Pass bowls of chopped dill and sour cream at the table.