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.
You can make your own coconut milk if you don't have any on hand. Blend together 1/4 cup coconut flaked with 1/2 cup water and milk, and then refrigerate overnight. Strain the next day for homemade coconut milk.
Per Serving: 248 calories;8.4 g fat;
41.7 g carbohydrates;
4.6 g protein;
4 mg cholesterol;
147 mg sodium.