First prepare the crust dough using the recipe for butter pie crust. Set the dough aside to rest in the refrigerator for at least an hour or up to overnight. (Since you only need enough dough for the equivalent of a single crust, you can half the recipe if you like...but I recommend simply freezing the remaining half for future use.)
On a lightly floured surface, roll the dough out into a circle about 12 inches in diameter. Using a sharp knife or pizza wheel, then cut the dough into very rough squares about 2 x 2 inches in size. Place the dough squares on a baking sheet (or half sheets) and refrigerate them for about 30 minutes. In the meantime, heat the oven to 400℉ and prepare the apples.
In the meantime, peel, core, and cut the apples into roughly ½ inch slices. Then place them in a bowl and add the lemon juice, brown sugar, cinnamon, and salt. Toss until the apples are evenly coated.
Place a 12 inch skillet over medium heat and add the butter. Once the butter has melted, add the apples and give them a good stir. Cover the pan and cook for about 15 minutes, stirring a couple of times along the way.
Stir the cornstarch into the apple cider and then pour the mixture over the apples in the skillet. Cook for a couple minutes more, or until slightly thickened.
Transfer the apples to a deep pie plate (or other baking dish with a minimum 1.5 quart capacity).
Arrange the chilled dough over the top of the apples, working around the pie plate in roughly concentric circles and slightly layering the edges of the squares like shingles on a roof.
Lightly beat the egg in a small bowl to make an egg wash. Brush the egg wash thoroughly over the top of the crust dough. Then sprinkle the white sugar across the entire surface.
Place the pie dish on a baking sheet (to catch any sticky drips), and bake for 15 minutes. Then remove the pandowdy from the oven. Using a large spoon, press down on the center of the crust and smoosh it into the fruit below until the juices begin to seep up. Repeat this motion with the spoon every few inches around the perimeter of the dish, about 4 or 5 times.
Put the pandowdy back in the oven and bake for about another 15 minutes, or until the crust is golden and the apple juices are bubbling up around the edges.
Serve warm or at room temperature, along with a generous scoop of good vanilla ice cream.