Thaw puff pastry in the refrigerator overnight. In a small bowl, mix blueberries, sugar, rum, lemon juice, salt and cornstarch together. In another bowl, mix together cream cheese, egg, powdered sugar and vanilla extract.
Preheat oven 375° F. Unwrap puff pastry and cut out 3 inch rounds. Place one teaspoon of blueberry mix, followed by one teaspoon of cheesecake mix on the pastry. Fold top over and crimp edges together with a fork. Make sure the edges are sealed well, otherwise the blueberry juices will ooze out when baking. Brush tops with egg wash (add one tsp. of water to the egg to make the egg wash), then make two small slits and sprinkle ¼ teaspoon sugar. Bake in the oven until golden brown, (about 20 to 25 minutes).