Pie: On a lightly floured piece of parchment, roll out dough into a 13-inch round, about 1/8 inch thick. Fit into a 9-inch pie dish. Using kitchen shears, trim to a 1-inch overhang; fold under. Crimp edges or decorate as desired. Lightly prick bottom of dough with a fork. Refrigerate 30 minutes.

Preheat oven to 375 degrees with a rack in lower third. Line dough with parchment; fill with dried beans or rice. Bake until edges begin to turn golden, 20 to 25 minutes. Remove beans or rice and parchment; bake until bottom and sides are golden brown, about 20 minutes more. Transfer to a wire rack; let cool completely. (Crust can be made up to a day in advance and covered tightly in foil.)

Combine granulated sugar, cornstarch, and salt in a medium saucepan. Whisk in coconut milk; cook over medium-high heat, stirring constantly, until bubbling and thick, about 7 minutes (about 2 minutes after it comes to a boil).

Whisk egg yolks in a large heatproof bowl until combined. Add coconut-milk mixture in a slow, steady stream, whisking until completely incorporated. Return mixture to pan; cook over medium heat, stirring constantly, until it returns to a boil, 1 to 2 minutes. Boil 1 minute more, whisking constantly.

Remove from heat; add butter, 1 tablespoon at a time, whisking until each piece melts before adding next. Whisk in coconut. Transfer pan to wire rack; let cool, whisking occasionally, 10 minutes. Pour custard into crust. Press plastic wrap directly on surface and refrigerate until custard is chilled and firm, at least 6 hours and up to 2 days.

Topping: Preheat oven to 350 degrees. Spread coconut in an even layer on a rimmed baking sheet and toast until golden, 8 to 10 minutes. Meanwhile, whisk cream to soft peaks. Add confectioners' sugar and vanilla; whisk until stiff peaks form. Spread whipped cream over custard. Serve immediately, or refrigerate, uncovered, up to 1 hour. Top with toasted coconut just before serving.