Heat oil in a soup pot over medium heat. Cook onion and carrot for 6 minutes, then add grated beets and cook for 5 minutes, stirring often.
Stir in tomato paste and garlic for 1 minute. Add broth, potatoes, cabbage, bay leaf, salt and pepper and bring to a simmer.
Cover partly and simmer for 25 to 30 minutes until potatoes, cabbage and beets are tender. Remove and discard bay leaf.
Stir in vinegar and chopped dill, then remove from the heat. Divide among 6 bowls and add 1 tablespoon sour cream to each serving.