Special equipment: A 9 1/2-inch deep-dish pie pan; pie weights, uncooked rice or dried beans
For the crust: Position the oven rack in the middle of the oven and preheat the oven to 350 degrees F. Coat the bottom and sides of a 9 1/2-inch deep-dish pie pan with nonstick cooking spray.
Roll the pie dough out, if needed, into a 10-inch round on a lightly-floured surface (or between two pieces of floured parchment paper). If the dough gets too warm, refrigerate it to firm up. Ease the dough into the prepared pie pan. Fold the overhanging dough under itself and crimp the edges as desired. Chill for 30 minutes.
Meanwhile, spread the pecans out on a baking sheet and bake until golden and toasted, 8 to 10 minutes. Let cool and separate 1/3 cup; coarsely chop the 1/3 cup pecans for garnish and set aside.
Put the pie pan on a baking sheet and line the chilled pie crust with foil or a large coffee filter and fill with pie weights. Bake until the edges are just barely golden, 15 to 18 minutes. Remove the foil with weights, then brush the bottom and edges of the dough all over with the egg white and continue baking until the bottom of the crust is also just barely golden, 8 to 10 minutes more. Transfer to a rack and cool completely. (Par-bake your pie shell up to 1 day in advance.)
For the pecan pie filling: Scatter the remaining 1 cup pecans in the bottom of the cooled pie shell. Whisk the corn syrup, brown sugar, butter, egg and 1/4 teaspoon salt together in a medium bowl. Pour the egg mixture over the pecans. Bake until just set, 20 to 25 minutes. Transfer the pie to a rack until cool to the touch.
For the apple pie filling: Meanwhile, peel, core and cut the apples into 1/4-inch slices. Toss with the granulated sugar and lemon juice in a medium bowl. Melt the butter in a large nonstick skillet over medium heat, then add the apples and cook, gently stirring occasionally, until the apples are soft but still hold their shape, about 10 minutes. Stir in the flour and cool completely. (Spread the mixture out on a baking sheet in a thin layer to help cool faster).
Pile the cooled apple mixture up and around the edge of the cooled pie (directly on top of the pecan filling) leaving about an 8-inch well in the center of the pie.
For the pumpkin pie filling: Whisk together the pumpkin, heavy cream, granulated sugar, egg and spice in a medium bowl. Pour the filling into the center of the pie (directly on top of the pecan filling) with the apples as a border.
Bake the pie until the pumpkin center is set with just a slight jiggle, 50 minutes to 1 hour. If the crust gets too brown, cover with foil or a pie crust shield. Let the pie cool completely on a rack.
Sprinkle the reserved 1/3 cup chopped pecans around the pie, snug up against the apple filling. Slice and serve with dollops of whipped cream.
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