Preheat oven to 400 F / 200 C.
Use two slightly different sized circular cookie cutters to cut as many circles from the puff pastry sheets as possible. I use 2-inch and 2.5-inch cutters. You can cut the smaller circles from one sheet, and the larger circles from the other sheet, to ensure that I make the same amount of each.
Transfer the smaller puff pastry circles to a parchment paper covered baking sheet. Then, add a heaping teaspoon of mincemeat in the middle of each circle.
Crack the egg into a small bowl, and use a small whisk or a fork to whisk the egg together to form a simple egg wash. With a pastry brush, brush some beaten egg on the edges of the pastry circles.
Cover each of the filled circles with one of the larger puff pastry circles. Gently fold the pastry around the filling.
Roll the leftover pastry into sheets to create additional pastries. Remember to create the same amount of smaller and larger pastry circles.
Use the tines of a fork to push down the edges of the pastry on each pie. This will allow the top pastry to fuse with the bottom pastry, and seals the edges.
Brush egg wash on top of the pastries, and use a sharp knife or a fork to poke a hole on top of each of them.
Bake the puff pastry mince pies for about 20 minutes, or until the pastries are golden brown on top. Then, remove them from the oven, allow them to cool down, and serve.