Bring chicken stock, beetroots, onion, carrot, garlic, lemon juice and sugar up to a simmer. Cook beets, loosely covered, until they are tender, about 40 minutes.
Purée until smooth, strain and cool to room temperature. Whisk in buttermilk and season. Stir in dill. Chill at least 4 hours before serving. Check seasoning, as soup may need more salt and pepper. Garnish with sour cream.