For the compote, in a small sauce pan add the blueberries, water, and white sugar to the pan, mix well and cook for 5 minutes on high and then 10 minutes on medium-low. Refrigerate for at least 1 hour.
Combine the whipped cream cheese and white sugar in a blender and blend for 15 seconds. Then add the heavy whipping cream, whole milk, and vanilla extract. Blend for an additional 15 seconds. Pour the milk mixture into a KitchenAid Ice Cream attachement and churn for 15 minutes (if using another ice cream unit follow manufacturers directions). During the last minute of churning add in one tablespoon of the blueberry compote. Once the ice cream is done begin scooping ice cream into a container. Fill half way and pour half of the compote into the container. Lightly mix and then add the remaining ice cream and blueberry compote and mix again. Eat immediately or freeze for up to one week.