In a small mixing bowl, combine the chopped pecans, brown sugar, corn syrup, melted butter, vanilla extract, and a pinch of salt. Stir everything together until the pecans are thoroughly coated and the sugar has partially dissolved into a thick, glossy syrup.
Unroll your pie crust on a lightly floured surface. Use a 2.5-inch or 3-inch round cookie cutter (or a wide-mouth glass) to cut out 12 circles. If you run out of space, gently gather the scraps, roll them out again, and cut the remaining circles.
Place the pastry circles into silicone mini muffin liners or directly onto a piece of parchment paper. Spoon about one tablespoon of the pecan mixture into the center of each circle. You can leave them as open-faced mini tarts or slightly pinch the edges upward to create a decorative rim.
Lightly brush the exposed edges of the pie crust with the whisked egg. This step is crucial for achieving that beautiful, golden-brown finish and a slight shine once they are cooked.
Preheat your air fryer to 330°F (165°C). Place the bites into the air fryer basket in a single layer, ensuring they are not touching. Cook for 8 to 10 minutes, or until the pastry is golden and the filling is bubbling. Remove carefully and let them cool for at least 5 minutes to allow the filling to set.