Apple-Cider Pie

To make the cake, lightly pour the flour into dry measuring cups; flatten with a knife. Combine 1/3 cup flour and iced water, stirring with a whisk, until smooth. Combine 1/2 cup flour and salt in a bowl; add margarine and blend with a blender or two knives until the mixture resembles coarse flour. Add the ice water mixture; stir with a fork until smooth. Divide the dough in half. Gently squeeze each half of the mixture into a 4-inch circle on thick plastic wrap and cover with extra plastic wrap. Roll out half of the dough, without removing the lid, into a 12-inch circle and refrigerate. Roll out the other half of the dough, without removing the lid, into an 11-inch circle; refrigerate.To make the filling, bring the cider to a boil in a large heavy saucepan over high heat. Cook until reduced to 1/2 cup (about 20 minutes). Allow to cool completely.Preheat the oven to 450 degrees.Combine the chilled cider, 1/3 cup sugar, cornstarch, lemon juice, vanilla, and pie spice in a large bowl. Slice each quarter apple crosswise into 1/4-inch thick slices. Add the apple slices to the cider mixture.Remove 1 sheet of plastic wrap from the 12-inch circle; place the dough on a 9-inch pie dish, smeared with cooking spray, so that the dough protrudes over the edges of the mold. Remove the top sheet of plastic wrap. Spoon the apple mixture onto the cake and lightly brush the edges of the cake with water. Remove 1 sheet of plastic wrap from the 11-inch circle and place on top of the apple mixture. Remove the top sheet of plastic wrap. Press the edges of the dough together; bend the edges and make a recess. With a sharp knife, cut 6 (1-inch) slits in the top of the dough. Mix the egg and 1 tablespoon of water. Brush the top and edges of the pie with the egg mixture and sprinkle with 1 tablespoon of sugar. Place the cake on a baking sheet and bake at 450 ° C for 15 minutes. Reduce the oven temperature to 350 ° (do not remove the cake from the oven) and bake for another 45 minutes or until golden brown. Cool on a wire rack.