1. Preheat oven to 400 degrees Fahrenheit.
2. Melt 1 tablespoon of the butter in a medium frying pan over medium heat. Add the thinly sliced apples to the pan and cook for 5 minutes, stirring occasionally.
3. In a small bowl, whisk together 1/3 cup of the brown sugar, the corn starch, and the cinnamon. Stir the mixture into the cooked apples in the pan and cook for an additional 3-5 minutes over medium heat until the apples are softened. Remove from the heat and set aside to cool to room temperature.
4. Unfold the puff pastry sheets and lay out on a flat 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 12 rectangles.
5. Arrange the puff pastry rectangles on a large, ungreased cookie sheet leaving at least 1 inch of space between each rectangle.
6. With a paring knife, lightly score a border around the entire edge of each pastry rectangle, about ¾ inch away from the edge. Use a fork to poke a few holes into the center area of each pastry. This process allows the outside border to rise and puff up while the center stays shorter and holds the filling.
7. Spoon the cooked apples onto each rectangle of pastry dough. Do not place the apple filling on the outside scored border of the pastry and do not add much of the liquid from the apples.
8. In a small bowl, use a fork to combine 2 tablespoons of butter, 3 tablespoons of brown sugar, and ¼ cup of flour. Mix until completely combined and crumbly in texture. Sprinkle the crumble topping over the apples.
9. In a small bowl, whisk the egg and water together. Use a pastry brush to apply a thin layer of the egg wash to the outside border of each pastry. Discard any leftover egg mixture.
10. Bake the pastries at 400 for 20 minutes, until they are golden brown.
11. In a small bowl, combine the powdered sugar and heavy cream; stir until smooth.
12. Cool the danishes on a baking rack for 10 minutes. Drizzle the frosting over the danishes.

Makes 12 servings.