Place beef, beets, tomatoes, potatoes, carrots, onion, and garlic in a slow cooker. Whisk together the beef broth, tomato paste, vinegar, brown sugar, dill weed, parsley, bay leaf, salt, and pepper. Pour mixture over the beef and vegetables, adding more broth to cover as needed.
Cover and cook on Low for 8 1/2 hours, or High for 4 hours.
Set heat to High, then stir in the shredded cabbage. Cover and continue cooking until the cabbage tender, about 30 minutes. Remove bay leaf. Serve in a bowl with a dollop of sour cream.