In a large soup pot, combine the beef broth, grated beets, diced potatoes, tomato paste, bay leaf, and a pinch of salt. Bring to a boil over medium-high heat, then reduce to low and simmer for 30 minutes.

While the broth simmers, heat a tablespoon of vegetable oil in a skillet over medium-low heat. Add grated carrots and diced onions, sautéing until softened, about 7 to 10 minutes. Stir in the minced garlic and cook for an additional 30 seconds.

Transfer the sautéed vegetables into the simmering pot of broth. Gently stir in the thinly sliced green cabbage and simmer for another 15 minutes.

Stir in the white vinegar, adjust seasoning with salt and pepper. Ladle into bowls and top with sour cream and a sprinkle of fresh dill.