Saute first 3 ingredients in hot oil in a Dutch oven until tender. Add cucumber and next 4 ingredients; bring to a boil. Reduce heat, and simmer 20 minutes or until tender. Cool.
Process soup in batches in a blender or food processor until smooth, stopping to scrape down sides. Pour through a fine wire-mesh strainer into a bowl. Cover and chill at least 4 hours.
Beat whipping cream at medium speed with an electric mixer until soft peaks form; fold in chopped mint. Serve soup with mint whipped cream.
Note: Vichyssoise may be frozen up to 1 month. Thaw in refrigerator overnight.