Make the gazpacho: Combine the tomatoes, cucumber, bread, garlic and vinegar in a large bowl; toss well. Working in batches, transfer to a blender and puree until smooth and thick, about 3 minutes. With the motor running, slowly add the olive oil and puree until incorporated. Add the sugar, 1 teaspoon salt, and pepper to taste and pulse to mix. Transfer to a large bowl, cover with plastic wrap and refrigerate until cold, about 1 hour.
Prepare the toppings: Rinse out the blender, then add the cilantro, olive oil, 1/4 cup water, the juice of 1/2 lemon and the sugar and puree until smooth. Season with salt.
Combine the cucumber, chile and the juice from the remaining 1/2 lemon in a bowl; season with salt and toss. Divide the soup among bowls. Drizzle with the cilantro oil and top with the cucumber-chile mixture.