To make the dough in a food processor, combine the flour, sugar and salt in a bowl. Add the butter and shortening and pulse until reduced to ½ inch pieces. Add the water a little at a time and pulse until the dough just begins to come together in a rough mass.
To make the dough by hand, combine the flour, sugar and salt in a bowl. Add the butter and shortening and toss to coat with the flour mixture. Using a pastry blender or 2 knives, cut the pieces of fat into the flour mixture until they are no larger than small peas. Distribute the water over the mixture and toss with a fork until the dough is evenly moist and begins to come together in a rough mass.
Transfer the dough to a work surface and shape into a 5-inch disk. Wrap with plastic wrap and refrigerate until well chilled, at least 2 hours.
Lightly dust a work surface and rolling pin with flour. Roll out the chilled dough into a 12-inch round a scant ¼ inch think. Lift and turn the dough several times as you roll to prevent s sticking, and dust the surface and rolling pin with additional flour as needed. Use a dough scraper or icing spatula to loosen the pastry if it sticks. Trim this dough into an 11-inch round. Slide a rimless baking sheet under the dough, cover with plastic wrap and refrigerate until needed.
Preheat an oven to 375 degrees.
To make the filling, set a 10-inch straight-sided, ovenproof fry pan, preferably cast iron, over medium heat and melt the butter. When it melts, sprinkle the sugar evenly over the butter and continue cooking until the sugar melts and turns amber colored, 3 to 4 minutes. Shake and swirl the pan frequently to redistribute the sugar for even melting and caramelization.
Arrange the apples, side up, in the caramel in a snug, even layer. Increase the heat to medium-high and cook until the apples are just tender about 15 minutes. The caramel will bubble up around the apples. Remove the pan from the heat.
Uncover the pastry round. When the bubbling has subsided, slide both hands under the pastry round and carefully place it on top of the apples, tucking in the edges and being careful not to burn your fingers. Bake until the crust is golden brown. About 30 minutes.
Transfer the pan to a wire rack and let cool for 6 minutes. Place a large flat serving plate upside down on top of the pan and invert the pan and plate together. Lift off the pan. Slice and serve warm with vanilla ice cream.
***It's really not essential to take this apple creation from your skillet - serve it from your cast iron and you'll feel authentic. Kinda romantic to be so casual about something so yummy.