Heat oil in a soup pot over medium-high heat. Saute garlic for 1 min. Add shredded beets, cabbage, carrots, and onion. (A food processor with a shredder attachment makes shredding go fast.) Saute for 10 min. stirring often.
Add soup stock. Bring to a boil and simmer for 15 minutes partially covered. Add remaining ingredients and simmer for another 15 minutes. Ladle into bowls. Garnish with sour cream. Serve with bread and sautéed beet greens.