In a small bowl, whisk together the whole egg with 2 teaspoons water, and set aside. On a lightly floured surface, roll out the piecrust to an 1/8-inch thickness. Transfer the crust to a 9-inch-round pie tin, and press it into the bottom edges and up the sides. Using scissors or a sharp knife, trim the crust so it overhangs the tin by about 1-inch. Turn overhanging dough under, forming a rim, and crimp the edges of the dough. Lightly prick the bottom of the dough with a fork; brush the rim with the beaten egg. Transfer crust to the freezer for 30 minutes.
Preheat the oven to 375 degrees F with the rack positioned in the lower third. Carefully line the crust with parchment paper, gently pressing the parchment into the edges of the crust weigh down parchment with dried beans or pie weights.
Bake until the edges of the crust begin to turn golden, about 15 minutes. Carefully remove the pie weights and the parchment paper. Continue to bake until crust is golden brown, 15 to 20 minutes more. Transfer the crust to a wire rack to cool completely.
Place 1/4 cup of the coconut on a silpat (a French nonstick baking mat) or parchment paper-lined sheet pan. Bake until light golden brown, about 5 minutes. Set aside.
To prepare the filling: Combine the cornstarch, 1/2 cup sugar, and salt in a medium saucepan. Whisk in milk, and cream of coconut. Set over medium heat, and cook, stirring constantly, until bubbling and thick, about 5 minutes.
Prepare an ice bath; set aside. In a medium bowl, whisk 4 egg yolks. Slowly whisk in hot milk mixture. Return mixture to saucepan, and continue cooking over medium heat, stirring constantly, until mixture returns to a boil, about 1 minute more. Remove from heat.
Stir in the vanilla and the butter. Fold in 1 cup of the coconut. Transfer custard to a medium bowl. Place over ice bath, and cover the surface of the custard with a piece of plastic wrap to prevent a skin from forming. Set aside.
Combine the 7 egg whites and the remaining 3/4 cup plus 2 tablespoons sugar in the heatproof bowl of an electric mixer. Set the bowl over a pan of simmering water. Whisk constantly until the sugar has dissolved and the whites are hot tot he touch, 5 to 7 minutes.
Transfer the bowl to the electric mixer fitted with the whisk attachment. Starting on low speed for 1 to 2 minutes and gradually increasing to high speed, beating until stiff, glossy peaks form, about 10 minutes.
Heat the broiler with the rack positioned in the lower third of the oven. Transfer the custard to the cooled crust. Using a large spatula, pile the meringue onto the center of the custard. Lightly spread the meringue over the custard, spreading the meringue so it touches the edges of the piecrust and sealing well to avoid shrinkage.
Transfer assembled pie to oven; broil until meringue turns golden brown, just a few seconds, watching constantly.
Transfer pie to a wire rack to cool at least 2 hours. Sprinkle with reserved 1/4 cup toasted coconut. Slice, and serve. Refrigerate any leftover pie.
2 1/2 cups all-purpose flour
1 teaspoon salt
1 teaspoon sugar
1 cup (2 sticks) unsalted butter, chilled, and cut into small pieces
Combine the flour, salt, and sugar in the bowl of a food processor. Pulse to combine. Add all of the butter, and pulse just until mixture resembles coarse meal; some of the pieces of butter should still be visible. Pulse in 1/4 cup ice water; the mixture should hold its shape when pinched together and will not form a ball in the processor. If necessary, add up to 1/4 cup more ice water.
Divide the dough in half. Turn each half onto a large piece of plastic wrap. Pat the dough into flat disks, and wrap tightly. Chill at least 1 hour or overnight. The piecrust may be stored in the refrigerator up to 2 days or frozen for weeks.
Yield: 2 (9-inch) crusts
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