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To make the crust: In a medium bowl, whisk together the flour, sugar, and salt.
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Cut the butter into 1/2" pieces and work it into the flour mixture with your fingers, a pastry blender, or a mixer. Don't be too thorough; the mixture should be unevenly crumbly with larger, walnut-sized chunks of butter among smaller, pea-sized ones.
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Make a well in the center of the flour and butter mixture. Separate the egg, then add the yolk to the well and set aside the white to use in the filling, leaving it at room temperature. Add 3 tablespoons (43g) of the water to the well, then whisk together the yolk and water.
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Using your hand or a fork, stir the egg and water into the flour mixture. Sprinkle in more water 1 teaspoon at a time, mixing it into the dough, until it's moist enough to hold together when squeezed and there are no remaining dry floury patches. (Don’t be tempted to add more than 3 teaspoons of additional water here or your crust will become tough.)
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Transfer the dough to a piece of plastic wrap or your favorite reusable wrap and pat it into a disk about 3/4" thick. Refrigerate the crust for at least 30 minutes or up to 2 days. If the dough chills longer than 30 minutes, allow it to rest at room temperature for 10 to 15 minutes before rolling, so the butter can soften a bit.
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To prebake the crust: On a lightly floured surface, roll the pie dough into a 12" circle about 1/8" thick. Transfer the dough to a 9" pie pan and trim the edges so they overhang the sides of the pan by an inch all around. Tuck the overhanging dough under itself, then flute or crimp the edge as desired. Place the lined pie pan in the refrigerator, uncovered, to chill for 1 hour.
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Toward the end of chilling, preheat the oven to 375°F with racks in the center and bottom positions.
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Once chilled, line the pie dough with foil or parchment and fill it with pie weights or dried beans. Bake on the bottom rack for 20 minutes. Remove it from the oven, and gently remove the foil or parchment with the weights or beans. Poke ("dock”) the bottom and sides with the tines of a fork to prevent puffing. Return the crust to the oven’s bottom rack for 10 to 15 more minutes, or until it's light golden brown all over. Remove the crust from the oven and transfer it to a wire rack to cool slightly while you prepare the filling.
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Reduce the oven temperature to 325°F.
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To make the filling: In a large bowl, whisk together the condensed milk and evaporated milk. Separate 2 of the eggs and add the yolks to the milk mixture; reserve the whites. Whisk in the remaining 3 whole eggs, vanilla, and almond extract; set aside.
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In a separate bowl, beat 3 egg whites (2 reserved from the previous step plus the additional white leftover from making the crust) until soft peaks form. Fold the beaten egg whites into the condensed milk mixture.
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Pass the filling through a strainer into a large bowl to remove any lumps. Use a flexible spatula to stir it through the strainer and scrape the underside.
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To assemble and bake: Pour the filling mixture into the pie crust. Bake on the middle rack for 50 to 55 minutes. The filling should look slightly domed, puffy, and deep golden brown. The edges will look set, but the center will still be jiggly. A digital thermometer inserted into the center of the pie should read between 175°F and 180°F.
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Remove the egg pie from the oven and place it on a wire rack to cool completely. Once cooled, refrigerate the pie, uncovered, for at least 6 hours, or preferably overnight. Resist the temptation to slice the pie before 6 hours, as it needs time to set up.
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Storage information: Store leftover egg pie in the refrigerator, covered, for up to 3 days.
