Preheat a grill or grill pan over high. In a food processor, pulse the first 5 ingredients until very finely chopped. Transfer to a large bowl.
In the processor (no need to clean the bowl), puree the tomatoes, vegetable juice, garlic, vinegar and 1/4 cup EVOO. Transfer to another large bowl.
Still working in the processor, blend the tomato mixture, bread and half the finely chopped vegetables until a thick soup forms. (You may need to work in batches.) Combine the pureed soup with the remaining chopped vegetables. Stir in the lemon juice and hot sauce; season with salt and pepper and transfer to a chilled tureen or large pot. Chill in the freezer, stirring occasionally, while you grill the shrimp.
Drizzle the shrimp with EVOO; season with Old Bay. Grill the lemon halves, cut side down, until lightly charred, 3 minutes, and grill the shrimp until opaque in the center, 3 to 4 minutes per side. Transfer the shrimp to a plate; douse with juice from the lemon halves. Divide the soup among small bowls. Garnish with the parsley and basil leaves; drizzle with EVOO. Serve the shrimp alongside for peeling and dipping.