Line a baking sheet with parchment paper. On a lightly floured surface, roll out the puff pastry with a floured rolling pin to 9 by 13 inches. Transfer to the lined baking sheet, cover with plastic wrap and refrigerate while preparing the pie filling.
Preheat the oven to 375 degrees F. Coat a 9-by-13-inch baking dish with nonstick cooking spray.
Melt the butter in a large skillet over medium heat. Add the celery and onions and cook, stirring occasionally, for about 5 minutes. Stir in the flour and cook 2 minutes more. Add the turkey, broth, half-and-half, Brussels sprouts, stuffing mix, salt and pepper and stir to combine. Cook until the sauce begins to thicken, 5 to 7 minutes. Remove from the heat.
Transfer the mixture to the prepared baking dish. Use a tablespoon to drop the cranberry sauce randomly over the mixture. Top with the puff pastry dough. Use a pastry brush to evenly coat the puff pastry with the egg. Using a paring knife, cut a few small vents in the center of the pie.
Bake until the puff pastry is golden brown, about 45 minutes. Let stand 5 minutes before cutting.