Preheat the oven to 375°F. Line a large baking sheet with parchment paper.
Unfold the puff pastry sheets and cut each sheet into 8 4-by-4 inch squares. Prick each square with a fork. Place half onto the prepared baking sheet.
Spread one tablespoon melted semi-sweet chocolate onto each square on the baking sheet, leaving a ½-inch border all around. Place a marshmallow in the center of each one.
Cover each square with one of the remaining squares of puff pastry and use a fork to crimp the edges together. Add one tablespoon of water to the beaten egg, then brush the egg wash over the tops of the poptarts.
Using a knife, cut three slits into the top piece of dough on each poptart. Bake until puffed and golden brown, rotating the baking sheet halfway through, 20-25 minutes.
Drizzle the poptarts with the remaining melted semi-sweet chocolate and the melted white chocolate, then sprinkle with graham cracker crumbs. Serve warm.