Peel, core and slice the apples into 1/4-inch thick wedges. Sprinkle lemon juice over the apples and toss to combine. Add the granulated and brown sugars to the apples and stir to coat. Cover the bowl with plastic wrap and set aside at room temperature while rolling out the dough.
Remove the dough from the refrigerator and allow it to rest at room temperature for 5 to 10 minutes before rolling.
Lightly flour a large piece of parchment paper. Roll the dough out on the parchment into a 10-inch circle for a 9-inch skillet (depending on the size of your skillet or casserole). If using a 13x9-inch pan roll the dough into a 14x10-inch rectangle. Using a pizza cutter or sharp knife, cut the dough into 2 1/2-inch squares. Transfer the parchment (with crust squares) to a baking sheet, cover loosely with plastic wrap and refrigerate for 30 minutes or until the dough is firm.
Preheat oven to 400°F. Lightly grease a 9 or 10-inch skillet or 13x9-inch casserole with butter. Set aside.
Pour the sugared apples and any accumulated juice into the prepared pan. Spread into an even layer.
In a 2-cup measure whisk together the apple cider, vanilla, cinnamon, nutmeg, cornstarch and salt. Pour over the apples then dot with 2 tablespoons of butter cut into 1/2-inch cubes.
Remove the dough squares from the refrigerator and place the pieces over the filling until mostly covered, overlapping as needed. Brush the beaten egg over the dough. Combine the cinnamon and sugar and sprinkle over the pieces of squares.
Bake for 30 to 35 minutes or until the juices are bubbling. Remove from the oven and gently press down on the dough using the back of a spoon or spatula until the juices come over the top of the crust. (You can also carefully tilt the pan and spoon the juices over the crust.) Repeat until most of the dough has been submerged or covered. Return the pan to the oven and continue baking until the crust is golden brown and glazed, about 15 minutes more.
Cool slightly and serve warm with ice cream if desired. The juices will thicken as the pandowdy cools.