In a blender, combine half each of the coarsely chopped green tomatoes, cucumber and onion with the avocado, jalapeño, garlic, lime juice and 1 cup of cold water and puree until smooth. Transfer the puree to a large bowl.
Add the remaining coarsely chopped green tomatoes, cucumber and onion to the blender along with the 2 tablespoons of mint, the cilantro and 1/4 cup of olive oil and pulse to a chunky puree. Add the puree to the bowl and stir well. Refrigerate the soup until well chilled, about 1 hour. Season the gazpacho with salt and pepper and ladle it into chilled bowls. Garnish the cold soup with the tomato wedges, diced cucumber, mint leaves and a drizzle of olive oil and serve.