In a large pot, heat the olive oil over medium heat. Add the chopped onion and sauté until translucent, about 5 minutes.
Stir in the grated beets, diced carrots, and diced potatoes. Cook for an additional 5 minutes, stirring occasionally.
Add the minced garlic and tomato paste, mixing well. Cook for another 2 minutes until fragrant.
Pour in the vegetable broth and bring the mixture to a boil.
Once boiling, reduce the heat to low and add the shredded cabbage, apple cider vinegar, and sugar. Season with salt and pepper.
Cover and simmer for about 30 minutes, or until the vegetables are tender.
Taste and adjust seasoning if necessary.
Serve hot, garnished with fresh dill and a dollop of sour cream if desired.