Preheat oven to 425 degrees.
Peel, core and chop the apples.
Combine cinnamon, sugar, nutmeg and bread crumbs in a large bowl. Toss the apples in the cinnamon mixture. Set aside.
Roll out the first puff pastry sheet into a large rectangle. Using a sharp knife or pizza cutter, cut into 4 equal squares. Use a pastry brush to brush each square with beaten egg. Place a few spoonfuls of the apple mixture in the center of each pastry square. Fold opposite corners in towards the middle, overlapping the corners over top of the apples.
Brush the outside of each dumpling with the egg wash. Repeat this process with the second puff pastry sheet, so that you have a total of 8 dumplings.
Coat an 8x8 baking dish with nonstick cooking spray. Arrange the dumplings in the baking dish.
Melt butter in a microwave safe dish. Combine the melted butter with the remaining streusel ingredients in a small bowl. Sprinkle over the tops of the dumplings. Bake in the preheated oven for 15 minutes. Lower the oven temperature to 350 degrees. Bake an additional 25-30 minutes.
Drizzle with caramel sauce, if desired.