Preheat the oven to 400 degrees F.
Unfold the thawed puff pastry sheets and lay them out on a flat, lightly floured surface. Use a knife to cut along the fold marks to create three equal sized strips. Cut each strip in thirds to make a total of 9 squares. Arrange the puff pastry squares on a parchment paper lined baking sheet with at least 1" between each pastry.
Using a round cookie cutter that is slightly smaller than each puff pastry square, lightly score a circle into the middle of each square without cutting all the way through. Use a sharp paring knife to cut the pastry from each corner of the square to the circle. With a fork, poke a few holes into the circles of each pastry to allow the outside border to rise and puff up while the center stays lower and holds the filling.
In a small bowl, combine the cream cheese, powdered sugar, and vanilla extract. Use a hand mixer to combine the ingredients until light and fluffy.
Add the filling to a piping bag and squeeze about 1½ tablespoons into the middle of each circle. Dip your finger in a small bowl of cold water, then take each corner of the puff pastry and fold it over to create a star shape.
Whisk together the egg and water in a small bowl, then use a pastry brush to brush all of the exposed areas of the puff pastry. Sprinkle a little coarse sugar on top of each pastry.
Bake in your preheated oven for 12-18 minutes, until golden brown.
While they are baking, combine the powdered sugar, vanilla extract, and a tablespoon of milk into a small bowl. Mix well—the consistency should be somewhat thick, but still thin enough to drizzle. If the consistency is too thick, add a little more milk. If too thin, add more powdered sugar.
Once the danishes are baked, remove from the oven and cool on a wire rack for 10 minutes before drizzling the glaze over the tops. Enjoy!