For the dough: Pulse the flour, granulated sugar and salt in a food processor until combined. Add the butter, and pulse until the mixture resembles yellow cornmeal mixed with pea-sized bits of butter. Add the egg, and pulse until the dough just comes together. (If the dough is very dry, add up to 1 tablespoon cold water.) Remove the dough, gently pat it into a round, wrap it in plastic wrap and refrigerate until thoroughly chilled, at least 1 hour and up to 2 days.
For the filling: Meanwhile, put the lemon juice in a medium bowl. Peel and core the apples; cut each in half, and cut each half into 4 wedges. Add the apples and granulated sugar to the bowl with the lemon juice, and toss.
Melt the butter over medium-high heat in a large skillet. Add the sugared apples, and cook, stirring, until the sugar dissolves and the mixture begins to simmer, about 2 minutes. Cover, reduce the heat to medium-low and cook until the apples soften and release most of their juices, about 7 minutes.
Strain the apples in a colander over a medium bowl to catch all the juice. Shake the colander to extract as much liquid as possible. Return the juice to the skillet, and simmer over medium heat until it has thickened and lightly caramelized, about 10 minutes.
Toss the apples with the thickened juice, cinnamon, allspice and nutmeg in a medium bowl. Set aside to cool completely. (The cooled filling can be refrigerated for up to 2 days or frozen for up to 6 months.)
For the streusel topping: Whisk the flour, sugar and salt together in a medium bowl. Add the butter, and rub it into the flour mixture with your fingers until it has been absorbed. Mix in the walnuts.
To assemble and bake the pie: Position a rack in the lower third of the oven, place a baking sheet on it and preheat the oven to 375 degrees F. On a lightly floured work surface, roll the dough disk into a 12-inch circle. Layer it between pieces of parchment or wax paper on a baking sheet, and refrigerate for at least 10 minutes.
Ease the dough round into a 9-inch pie pan. Fold and crimp the edges as desired. Add the filling, and spread it out to make an even layer. Squeeze handfuls of the streusel topping, and drop clumps of it on top of the filling to completely cover it.
Bake the pie on the preheated baking sheet until the crust and streusel are golden brown, 50 to 60 minutes. Let it cool on a cooling rack for at least 3 hours before serving. (The pie keeps well, covered, at room temperature for 24 hours or refrigerated for up to 4 days.)
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