In a large stock pot, add the olive oil to the pot and bring the heat to medium high. Add the onion and garlic and saute with rosemary sprig, bay leaves, and basil for 2-4 minutes. Add the butter, then the rest of the vegetables (carrots, parsnip, leeks, celery, tomatoes, and mushrooms). Cook for 8-10 minutes, stirring occasionally.
Once your vegetables have softened, add the tomato paste along with the grated horseradish and then season with salt and pepper. Stir until the paste is evenly distributed throughout the soup, stirring until it turns a deep red color (about 3 minutes).
Add the red wine and stir into the vegetable base, removing any sticking vegetables from the bottom of the stock pot. Cook for about 3 minutes, reducing the wine by at least half.
Add the cut leftover prime rib beef portions and the first 6 cup portion of beef broth. Stir to combine, cover, and allow to simmer on low for about 70-75 minutes.
While the soup is simmering (about 40-45 minutes into the simmering period), bring a second sauce pan with 3 cups of the broth to a boil. Add the barley and cook at a low boil over medium heat until al dente, about 20 minutes. The liquid will not be absorbed completely, do not drain off excess broth.
When the soup has simmered for about 70-75 minutes, and the barley has cooked, combine the barley and broth with the soup. Stir, cover and simmer for an additional 10-12 minutes or until the barley is tender.
Remove the soup from heat. Stir in the red wine vinegar and serve immediately. Garnish with additional chopped basil (or parsley), if desired.