In a food processor, combine the flour, sugar and salt in the bowl and pulse to blend. Add the butter and the shortening and pulse until reduced to 1/2-inch (12mm) pieces. Add the water a little at a time and pulse until the dough just begins to come together in a rough mass.

Transfer the dough to a work surface and shape into a 5-inch (13 cm) disk. Wrap with plastic and refrigerate until well chilled, at least 2 hours.

Lightly dust a work surface and a rolling pin with flour. Roll out the chilled dough into a 12-inch (30 cm) round, a scant 1/4 inch (6 mm) thick. Lift and turn the dough several times as you roll to prevent sticking, and dust the surface and the rolling pin with additional flour as needed. Use a dough scraper or icing spatula to loosen the pastry if it sticks. Trim the dough into an 11-inch (28 cm) round.

Slide a rimless baking sheet under the dough, cover with plastic wrap and refrigerate until needed. Preheat an oven to 375F° (190C°).

To make the filling, set the tarte Tatin pan over medium heat and heat 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, core side up, in the caramel in a snug, even layer. Increase the heat to medium-high and cook until the apples are 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 5 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.