2) To make the pastry in a food processor, with the machine fitted with the metal blade, pulse the flour, sugar and salt until combined. Add the butter and pulse until it resembles yellow corn meal mixed with bean size bits of butter, about 10 times. Add the egg and pulse 1 to 2 times; don't let the dough form into a ball in the machine. (If the dough is very dry add up to a tbsp more of cold water.) Remove the bowl from the machine, remove the blade, and bring the dough together by hand.
3) Form the dough into a disk, wrap in clingfilm and refrigerate until thoroughly chilled, at least 1 hr.
To make the filling:
1) Put the lemon juice in a medium bowl. Peel, halve, and core the apples. Cut each half into 4 wedges. Toss the apple with the lemon juice. Add the sugar and toss to combine evenly.
2) In a large skillet, melt the butter over a medium-high heat. Add the apples, and cook, stirring, until the sugar dissolves and the mixture begins to simmer, about 2 mins. Cover, reduce the heat to medium-low, and cook until the apples soften and release most of their juices, about 7 mins.
3) Strain the apples in a colander over a medium bowl to catch all the juice. Shake the colander to get as much liquid as possible. Return the juices to the skillet, and simmer over a medium heat until thickened and lightly caramelized, about 10 mins.
4) In a medium bowl, toss the apples with the reduced juice and spices. Set aside to cool completely. (This filling can be made up to 2 days ahead and refrigerated or frozen for up to 6 months.)
To make the pie:
1) Cut the dough in half. On a lightly floured surface, roll each half of dough into a disc about 28 to 30cm wide. Layer the dough between pieces of parchment or wax paper on a baking sheet, and refrigerate for at least 10 mins. Place a rack in the lower third of the oven and preheat the oven to 190C/Gas 5.
2) Line the bottom of a 23cm pie pan with one of the discs of dough, and trim it so it lays about 1cm beyond the edge of the pan. Put the apple filling in the pan and mound it slightly in the centre. Brush the top edges of the dough with the egg.
3) Place the second disc of dough over the top. Fold the top layer of dough under the edge of the bottom layer and press the edges together to form a seal. Flute the edge as desired. Brush the surface of the dough with egg and then sprinkle with sugar. Pierce the top of the dough in several places to allow steam to escape while baking. Refrigerate for at least 15 mins.
4) Bake the pie on a baking sheet until the crust is golden, about 50 mins. Cool on a rack before serving. The pie keeps well at room temperature (covered) for 24 hours, or refrigerated for up to 4 days.
Cook's Note: You may freeze the uncooked pie, but don't brush it with egg or dust it with sugar beforehand. Place the pie in the freezer for 30 mins, to harden it slightly, and then double wrap it with cling film. Freeze for up to 6 months. When ready to bake, unwrap the pie and brush it with egg and sprinkle with sugar. Bake, from the frozen state, until golden brown, about 1 hr and 10 mins.