Sauté base vegetables: In a large pot, heat olive oil over medium heat. Add onion and sauté for 3–4 minutes until soft. Add garlic and stir for another 30 seconds.
Add beets and tomato paste: Stir in the grated beets and tomato paste. Cook for 5 minutes, letting the mixture caramelize slightly.
Simmer potatoes and cabbage: Add cubed potatoes, shredded cabbage, and water. Stir well. Bring to a boil, then reduce to a simmer.
Season and cook: Add salt, pepper, and sugar if using. Let simmer uncovered for 30–40 minutes, until vegetables are tender and flavors meld.
Balance the flavor: Right before serving, add vinegar to taste for a subtle tang.
Serve hot: Ladle into bowls. Top each serving with a dollop of sour cream and a sprinkle of chopped fresh dill.