Thaw the puff pastry according to the package directions, then keep it refrigerated until ready to use. Melt the butter in a skillet over medium heat and add the diced apples, brown sugar, cinnamon, lemon juice, cornstarch, and salt.

Cook the apple mixture for 5–8 minutes, stirring frequently, until the apples begin to soften and the juices thicken. Do not cook the apples until mushy because they will continue cooking inside the pies.

Set the filling aside to cool until it is no longer hot and has a slightly thick consistency. Lightly grease a standard 12-cup muffin tin and preheat the oven to 400°F/200°C.

Cut each puff pastry sheet into six approximately 4-inch squares, creating 12 squares total. Press one square into each muffin cup, leaving the corners extending above the rim.

Spoon the cooled apple mixture into the pastry cups, taking care not to overfill them. Fold the pastry corners loosely toward the center, or leave them open for a rustic pie-cup appearance.

Whisk the egg with the water to make an egg wash and brush it over the exposed pastry. Sprinkle the tops with coarse sugar if desired, and refrigerate the assembled tin for 10–15 minutes if the pastry has softened.

Bake the mini pies at 400°F/200°C for 20–25 minutes, until the puff pastry is deeply golden, well risen, and the apple filling is bubbling. Rotate the pan only if your oven browns unevenly.

Let the pies cool in the muffin tin for 5–10 minutes before transferring them to a wire rack. Dust with confectioners’ sugar if desired and serve warm.