In a 3-quart saucier or saucepan, melt butter over medium heat until foaming. Add leek and garlic scapes and cook, stirring often, until softened but not browned, about 6 minutes.
Add vegetable stock (or water) along with the potato and bring to a simmer over medium-high heat. Cook at a simmer until potato is tender, about 5 minutes. Remove from heat and let cool until warm (to speed up cooling, you can set the pot in a larger mixing bowl filled with an ice bath).
Transfer broth and solids to a blender. Starting at the lowest speed and gradually increasing to high, blend until solids are thoroughly pureed and soup has a smooth texture. Turn off blender, add mint and parsley, then blend again until herbs are fully incorporated into soup. Return to saucier and whisk in additional stock or water until soup is pourable yet thick enough to lightly coat a spoon (about 1 to 2 additional cups). Season with salt.
Serve soup warmed, at room temperature, or chilled, garnishing with a dollop of sour cream, crème fraîche, or a generous drizzle of buttermilk.
