Heat oven to 375°
In a small bowl combine the brown sugar, cinnamon, and the pecans, and apple pieces.
On a lightly floured surface, roll out the puff-pastry sheets, measure 9-by-6-inch piece, and cut with pastry wheel or knife.
Place the cored apple onto the middle of the puff-pastry, and spoon the brown sugar mixture into the center of the cored apple.
Lightly brush the beaten egg 1" around the edge of the puff pastry. Then gather the corners and pull up to the center, and twist or fold at the top to seal the apple. You can cut any excess off the top with scissors.
Put them into a baking dish, but first chill the dumplings for 15-20 minutes. Then brush the apple with the beaten egg (this will give it a shiny golden brown finish), and sprinkle the raw sugar on the top.
Bake the dumplings 25-30 minutes, or until golden brown. (cooking times can vary) Cut into halves, and place into bowl. Add vanilla ice cream, whipped cream, drizzle caramel syrup, and a few sprinkles for some color. :-)