Serve with goat cheese–topped toasted baguette slices, if desired.
Process 2-inch pieces of peach and cucumber, 1 cup ice water, scallion, coarsely chopped jalapeño, salt, and sugar in blender until completely smooth, about 2 minutes. Strain soup through fine-mesh strainer into large bowl, using back of ladle or rubber spatula to press soup through. Whisk in yogurt, 2 tablespoons vinegar, pepper, and remaining 1 cup ice water. Cover and refrigerate to meld flavors, at least 20 minutes or up to 12 hours.
Meanwhile, combine ¾-inch pieces of peach and cucumber, finely chopped jalapeño, and remaining 1 tablespoon vinegar in bowl.
Stir mint into soup and season with salt and pepper to taste. Serve cold, topping individual portions with peach-cucumber salad and shredded mint and drizzling with oil.
