Heat the olive oil in a large pot over medium heat. Add the scallions and potato and cook until the scallions begin to wilt, about 2 minutes. Stir in the beef broth. Pour the beet juices into a measuring cup and add enough water to make 2 1/2 cups. Add to the pot; season with 1/2 teaspoon salt and a few grinds of pepper. Bring to a simmer and cook until the potato is just tender, 8 to 10 minutes.
Meanwhile, mix the sour cream and horseradish in a small bowl; season with salt and pepper. Toast the bread until crisp. Brush the bread lightly with olive oil and season with salt and pepper.
Stir the vinegar and cubed beets into the soup; simmer until the beets are heated through, about 2 minutes. Stir in the dill.
Spread half of the horseradish sour cream on the toasts and top with the roast beef. Sprinkle with dill and season with salt and pepper. Divide the soup among bowls and top with the remaining horseradish sour cream and more dill. Serve with the roast beef toasts.