Place all the ingredients into the blender, withholding the oil for later. Add 1/2 teaspoon of salt and 1/4 teaspoon pepper. Then cover and puree until very smooth.
Then open the vent (on the lid of the blender) and slowly pour the olive oil into the gazpacho while the blender is running to emulsify. Taste for salt and pepper and add more if needed. Refrigerate until ready to serve.
Mix the diced watermelon, cantaloupe and cucumber in a bowl. Pour the watermelon gazpacho base into 4 serving bowls, then spoon the diced toppings into the middle of each bowl. Top with a mint sprig and a sprinkle of asiago shavings.