Dice 2 tomatoes, and set aside for garnish; core and quarter remaining tomatoes. In two batches, puree quartered tomatoes with tomato juice, vinegar, and 1 tablespoon oil in a blender until very smooth, about 3 minutes; transfer to a large bowl. Season gazpacho with salt and pepper, and refrigerate until ready to serve (up to 1 day).
Drizzle cut sides of bread with remaining 2 tablespoons oil. Layer prosciutto and mozzarella on bottom half of bread; season with salt and pepper. Close sandwich, and cut into quarters. Garnish soup with diced tomatoes, and drizzle with oil; serve sandwiches alongside.
For the most vibrant, flavorful version of this Spanish tomato soup, seek out the ripest local tomatoes at your market they should be deep red and aromatic.