Preheat the oven to 400 degrees F.
Unfold the thawed sheets of puff pastry and lay 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 half to make a total of 6 rectangles, then arrange them on a parchment paper lined baking sheet with at least 1" between.
Using a paring knife, lightly score a line about 1/2" inside of each rectangle. (Don't cut all the way through the pastry) Poke a few holes into the center of each pastry with a fork 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 the vanilla extract. Use a hand mixer and combine the ingredients until light and fluffy, then add to a piping bag and cut off the tip.
In a separate small bowl, combine the pumpkin puree, brown sugar, pumpkin pie spice, and cinnamon. Mix well, then add the mixture to a piping bag and cut the tip to about the width of a pencil.
Squeeze about 1½ tablespoons of cream cheese and pumpkin the filling into the middle of each rectangle. Use a toothpick to swirl the mixtures together.
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 about 15-18 minutes, or until the Danish are 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!