Peel and core the apples; slice 1/4 inch thick. Transfer to a bowl and toss with the granulated sugar and lemon juice. Melt 4 tablespoons butter in a large skillet over medium-high heat. Add the apples and cook, stirring occasionally, until softened, 12 to 15 minutes. Add the flour, cinnamon and salt and stir until the juices thicken, about 2 minutes. Remove from the heat and let cool completely. (The filling can be made up to 2 days ahead; cover and refrigerate.)
Roll out 1 disk of dough into a 12-inch round on a lightly floured surface. Ease into a 9-inch pie plate. Add the cooled filling and dot with the remaining 2 tablespoons butter.
Roll out the remaining disk of dough into a 12-inch round. Lay the dough over the filling and press the two crusts together around the edges. Fold the overhanging dough under itself and crimp with your fingers. Brush the top crust with the beaten egg and sprinkle with coarse sugar. Cut a few slits in the top crust to let steam escape. Chill 1 hour.
Put a baking sheet on the bottom oven rack and preheat to 425 degrees F for at least 30 minutes. Put the pie directly on the hot baking sheet and reduce the oven temperature to 375 degrees F; bake until the pie is golden and the filling is bubbly, 1 hour to 1 hour, 10 minutes, rotating the pie as needed. (Cover the edges with foil if they are browning too quickly.) Transfer to a rack and let cool until the filling is set, about 3 hours.
Pulse the flour, shortening, sugar, vinegar and salt in a food processor until it looks like fine meal. Add the butter and pulse until it is in pea-size pieces. Sprinkle in 1/4 cup ice water and pulse until the dough begins to come together. Pinch the dough with your fingers; if it doesn't hold together, add up to 4 more tablespoons ice water, 1 tablespoon at a time, and pulse again.
Divide the dough between 2 sheets of plastic wrap and pat each into a disk. Wrap tightly and refrigerate until firm, at least 1 hour or preferably overnight, or freeze up to 2 months.
Recipe courtesy Food Network Magazine