Whisk the flour, sugar, and salt together in a medium bowl. Rub 1/4 cup of the butter into the dry ingredients with your fingers until completely absorbed. Then rub in the remaining butter until it resembles cornmeal mixed with pea-size bits of butter. (If it gets warm and sticky, refrigerate it to chill.)
Beat the egg with 3 tablespoons of the water; then drizzle it evenly over the dough. Lightly stir the dough together with a fork. (The dough should just hold together when you squeeze it together, with some dry crumbly bits.) If the dough is really dry, sprinkle it with the final tablespoon of water. (To make the dough in food processor, see below.)
Divide the dough in half and wrap each half in plastic wrap and shape into disks. Refrigerate at least 1 hour or up to 2 days. (The dough can be frozen for 2 months. Defrost dough in the fridge overnight.)
Finely grate the lemon zest and set aside. Peel, core and then slice both the apple and pear into 1/2-inch slices. Squeeze the lemon juice over the fruit, then toss fruit with the sugar, cinnamon, ginger, salt and nutmeg.
Melt the butter in a large skillet over medium-high heat. Add the fruit and cook, stirring until the sugar dissolves and juices simmer, about 2 minutes. Reduce heat to medium, and cook, uncovered, until the fruit softens and the juices evaporate some, about 10 minutes. Evenly mix the flour into the fruit; then cook about a minute more to thicken the juices slightly. Stir in the vanilla and lemon zest; and remove from the heat. (The filling should resemble a tight compote.) Cool completely.
Form the pie:
Lightly dust the work surface with flour. Roll a disk of dough into an 11 to 12-inch circle. Transfer the dough to 9-inch glass pie pan (see photo), trimming so it hangs about 1/2-inch over the edge of the pan. Fill the crust with the prepared fruit so it mounds slightly in the center. Roll the remaining dough into a 12-inch circle. Brush the rim of the crust with some of the egg.
Roll the dough onto the rolling pin and unroll it over the fruit so it hangs over the edge of the pie pan by about 1/2-inch. Trim crust if needed, reserving the scraps for decorations or for patching, if needed. Fold the top crust edge under the bottom one, then press the edges together to seal. Cut trimmed scraps into designs if desired and set aside. Flute the crust by pressing a finger into the crust against the other hand's index finger and thumb to make an even impression. Repeat every 1/2 inch around the pie to create a ruffled edge (see photo). Refrigerate the pie for at least 20 minutes.
Meanwhile, place a rack in the lower third of the oven and heat to 425 degrees F.
Brush pie with egg and place cut dough designs on top if desired. Brush again and sprinkle with sugar. Cut 6 to 8 small steam vents into the top of the dough. Place pie on a baking sheet and cook for 15 minutes, then reduce the temperature to 375 degrees F. Bake until the crust (both top and bottom) is golden brown, about 50 minutes more. If the edges begin to brown too quickly, cut a pie shield out of a piece of aluminum foil and cover the edges (see photo). Cool on a rack.
Serve pie warm or at room temperature with whipped or ice cream. Keep pie, covered, at room temperature for a day, or refrigerate for up to four.
Cook's Note: To make the dough in a food processor: Pulse the flour, sugar, and salt together in the bowl of the processor. Add a 1/4 cup of the butter and pulse until it resembles fine cornmeal. Add the remaining butter and pulse until the butter is in small pea-sized pieces. Beat the egg and 3 tablespoons of the water together, add and pulse 1 to 2 times, but don't let the dough form into a ball in the machine (see photo). If the dough is dry add pulse in the other tablespoon of water. Remove the blade and turn dough onto a large, sheet of plastic film. Use the sides of the film to bring the dough together, then wrap tightly to chill.
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