Cook carrots, covered, in the 1 cup water in a medium saucepan for 15 to 20 minutes or until very tender. Cool slightly; transfer undrained carrots to a blender container or food processor bowl. Add orange peel, the 1 tablespoon dill, salt, and white pepper. Cover and blend or process until mixture is smooth.
Add additional water (about 2 cups) 1/2 cup at a time, blending or processing until mixture reaches desired soup consistency. Cover and chill 2 to 24 hours.
To serve, divide soup mixture in half. Return half to a saucepan and heat just until simmering. Remove from heat. Stir in cream. Divide hot soup among small cups, filling half full. Carefully spoon cold soup atop. Top with chopped beets or snipped fresh dill. Serve at once. Makes about 12 appetizer servings.