Heat the vegetable oil in a saucepan over medium heat. Cook the peaches in the hot oil until softened, about 5 minutes. Stir the brown sugar and vegetable broth with the peaches; continue cooking until the sugar dissolves, 10 to 15 minutes; remove from heat and allow to cool.
Stir the coconut milk and chopped mint through the peach mixture.
Pour the soup into a blender, cover, and puree until smooth. Pour the pureed soup into a large bowl, stir the peach yogurt thoroughly into the soup, and refrigerate until completely chilled. Top each serving with 1 tablespoon vanilla yogurt and a fresh mint leaf to serve.
