In a medium mixing bowl, combine the pumpkin puree, brown sugar, pumpkin pie spice, and vanilla extract. Stir until the mixture is smooth and the sugar is fully incorporated. Set this aside while you prepare the dough.
Unroll the refrigerated pie crusts onto a lightly floured surface. Use a 3.5-inch or 4-inch round cookie cutter (or a large glass) to cut out circles. You should be able to get about 8 to 10 circles by re-rolling the scraps as needed.
Place about 1 to 1.5 tablespoons of the pumpkin filling into the center of half of the dough circles. Be careful not to overfill, as the filling might leak out during cooking. Whisk the egg and water together in a small bowl to create an egg wash.
Lightly brush the edges of the filled circles with the egg wash. Place a second dough circle on top of the filling. Use a fork to crimp the edges together tightly, sealing the pie. Use a sharp knife to cut a small “X” or a few slits in the top of each pie to allow steam to escape.
Preheat your air fryer to 350°F (175°C). Lightly grease the air fryer basket or line it with perforated parchment paper. Brush the tops of the pies with the remaining egg wash and sprinkle with coarse sugar. Place the pies in the basket in a single layer, ensuring they don’t touch. Cook for 8 to 10 minutes, or until the crust is golden brown and flaky.