Begin by preparing the vegetables. Peel the beets, carrot, and potato. Chop the beets and potato into small cubes, grate the carrot, and finely chop the onion. Slice the cabbage into thin strips.
In a large pot, heat the vegetable oil over medium heat. Add the onions and sauté for about 5 minutes, or until they are translucent.
Stir in the grated carrot and cook for an additional 3-5 minutes.
Add the cubed beets and potato to the pot, stirring to combine. Cook for another 5-7 minutes, allowing the vegetables to slightly soften.
Pour in the broth and bring the mixture to a boil. Then, reduce the heat to a simmer.
Add the tomato paste, vinegar, and sliced cabbage. Season with salt and pepper. Simmer for about 30 minutes, or until all vegetables are tender.
Taste and adjust seasoning if necessary. Remove from heat and let it sit for a few minutes.
Serve hot, garnished with fresh dill and a dollop of sour cream on each bowl. Pair with crusty bread for a complete meal.