Preheat oven to 400°.
Use a pastry blender or fork to combine the flour, sugar, salt and shortening (butter). **The original recipe calls for shortening. I have made it with butter, same measurements. It works, although this pastry is very sticky and hard to work with. Totally worth it, because that is what makes this pastry so light and flaky. I do feel that the butter made it even more sticky and harder to work with. Separate one egg. Set the egg white aside. Place the yolk into a measuring cup. Add enough milk to make it ⅔ cup. Whisk with a fork or small whisk to combine the yolk and milk.
Add the milk and yolk to the crumbly flour mixture. Use a fork to combine it all without overworking it.
Split the dough in half. Lay out a sheet of plastic wrap about the size of whatever large baking sheet you are using. Spread the first half of the dough over the center of the plastic wrap. Lay another sheet of plastic wrap over the top. Roll it out while occasionally straightening out the plastic wrap.
Carefully peel the top sheet of plastic off. Lift the bottom two corners of the underneath sheet of plastic. Lift and turn it over onto the baking sheet. Carefully peel off the bottom sheet of plastic wrap, which is now on top.
Pile the peeled and sliced apples over the rolled-out pastry. Leave about an inch around the edge. Combine the sugar and cinnamon. Pour evenly over the top of the apples.
Roll out the other half of the dough like the first half, between two sheets of plastic wrap. Peel the top sheet off, then lift and turn onto the top of the apples. Carefully pull the second sheet of plastic off. **This dough is very soft, sticky and fragile. Do not stress too much about it. I rarely do this perfectly. Just piece it together and do the best you can. It will still bake and turn out delicious. Roll up and pinch the edges closed.
Pour the egg white into a larger bowl so you are able to use electric beaters to whip it up till white and stiff. (You could also do this by hand with a whisk, but it will take longer.)
Spread the whipped egg white over the top. Be careful not to break down the whites too much. I use the back of a spoon to carefully spread it.
Bake at 400° for 40 minutes.
Make a glaze by whisking the powdered sugar and lemon juice together. Drizzle it over the top while still hot. Let it cool for at least 30-60 minutes before serving in squares.