Combine beets and vinegar in a small frying pan over low heat. Cook, while stirring, until soft, about 15 minutes.
Heat oil in a large frying pan over low heat while beets are cooking. Add onion and stir for 2 minutes. Add carrots and cook, stirring occasionally, until soft, about 10 minutes. Set aside.
In the meantime, bring water to a simmer in a large saucepan. Add cabbage and lentils. Cook for 10 minutes. Add potatoes and cook for 10 minutes more. Stir in cooked beets and onion-carrot mixture. Season with salt and pepper. Add tomato paste and simmer until all vegetables are tender, about 10 minutes more. Serve with sour cream and dill."