Begin by peeling and grating the beets, then set them aside. Feel free to wear gloves to avoid staining your hands.
In a large pot or Dutch oven, heat the vegetable oil over medium heat. Add the chopped onion and sauté for about 5 minutes until it becomes translucent.
Stir in the grated carrot and continue to sauté for another 3-4 minutes.
Add the diced potatoes, sliced cabbage, and grated beets to the pot. Mix well and cook for an additional 5 minutes, stirring occasionally.
Pour in the broth, add the tomato paste, vinegar, and sugar. Bring the mixture to a boil, then reduce the heat and let it simmer for about 30 minutes, or until the beets and vegetables are tender.
Season the borscht with salt and pepper to taste. Adjust the vinegar and sugar as needed to balance the flavors.
Remove the pot from heat and let it sit for a few minutes to cool slightly.
Serve the borscht hot, garnished with fresh dill and a dollop of sour cream in each bowl.