Combine 1/4 cup cucumber, 2 tablespoons basil, 2 tablespoons onion and 2 teaspoons oil in a small bowl.
Working in batches, puree tomatoes, vinegar, salt, pepper and remaining cucumber, basil and onion in a food processor or blender. Serve immediately or refrigerate. Garnish with diced cucumber-onion mixture. Drizzle each serving with 1 teaspoon olive oil (optional).