Apple Tarte Tatin
In a mixing bowl, combine the flour and the salt. Quickly cut the butter into the flour, using a pastry blender or your fingers, until the butter pieces are the size of large peas. (Alternatively, cut the butter into the flour by pulsing it 8 to 10 times in a food processor, being careful not to overheat or to overmix the butter.) Dump the mixture out onto a clean surface and make a well in the center of the flour. Pour the ice water into the well. Using just your fingertips and working quickly, combine the flour mixture and the water. Work just until the water is absorbed. The dough will be ragged but should hold together when you squeeze it. If it seems dry, sprinkle in a few more drops of water. Form the dough into a log shape about 8 inches long, parallel to the edge of your work surface. With the heel of your hand, push down and away from you all along the line of dough. With a pastry scraper, gather up the dough, shape it back into a log, and repeat the smearing action. This technique, known as fraisage, will form sheets of butter in the dough, creating a light crust almost like puff pastry. With the pastry scraper, gather the dough up into a ball; it is fine if the dough does not come together completely at this time. Wrap the dough well in plastic wrap, flatten it a bit, and let it rest in the refrigerator for at least 30 minutes before rolling. The dough will keep in the refrigerator for up to 2 days. You can also freeze the dough, well wrapped; allow it to defrost for 1 day in the refrigerator before using it.
On a lightly floured surface, roll the rested dough into a 12- to 13-inch circle. Trim rough edges, if necessary, to maintain a round shape. Transfer the dough to a sheet pan, cover with plastic wrap, and refrigerate. Preheat the oven to 400°F. Toss the apples with the cinnamon and ¼ cup of the sugar. In a 10-inch cast-iron pan or other heavy-based, ovenproof skillet, heat the remaining ¾ cup sugar over medium-high heat, stirring to remove lumps, until the sugar has turned a dark amber. (Check for color by drizzling some onto a white plate.) Add the butter and stir it into the caramel until melted and homogenous; be careful as the butter will make the caramel sputter. Remove the pan from the heat. Beginning on the outside of the pan, carefully set the apple quarters in the caramel, arranging them in a circle so that they all face the same way. Set them very close to one another, trying to get as many as possible into the pan. Fill in the center of the pan with as many remaining apples as will fit. Put the pan back on the heat and allow the apples to cook on top of the stove for about 5 minutes. Remove the pan from the heat. Place the cold pie dough over the apples, tucking the edges into the sides of the pan. Work carefully so as not to burn your fingers but also quickly so as not to melt the dough. Brush the dough with the beaten egg and immediately put the pan in the oven. Bake for 15 minutes. Lower the temperature to 375°F and bake until the crust is nicely browned and the caramel is bubbling around the edges of the pan, another 15 minutes. Carefully remove the tart from the oven and cool it on a rack for about 20 minutes. Invert the tart onto a serving platter. If any apples stick to the underside of the pan, simply return them to their rightful spot on the tart.
Apple Tarte Tatin is best served right away, but it can stand at room temperature and be reheated briefly in the oven. Serve with whipped cream or vanilla ice cream.