On a lightly floured work surface, roll pate brisee to 1/8-inch thickness. Fit into a 9-inch pie dish. Trim edges, leaving a 1-inch overhang. Tuck overhang under dough so edges are flush with rim, and crimp edges. Lightly prick bottom of dough with a fork. Refrigerate for 30 minutes.
Preheat oven to 375 degrees. Line dough with parchment, and fill with pie weights or dried beans. Bake until edges begin to turn gold, 15 to 18 minutes. Carefully remove weights and parchment. Bake until bottom and sides are golden brown, 12 to 15 minutes. Let cool completely on a wire rack. (Crust can be stored overnight.)
Combine granulated sugar, cornstarch, and salt in a medium saucepan. Whisk in milk and vanilla seeds, and cook over medium-high heat, stirring constantly, until bubbling and thick, about 7 minutes (about 2 minutes after it comes to a boil).
Whisk yolks in a medium bowl until combined. Pour in milk mixture in a slow, steady stream, whisking until completely incorporated. Return the mixture to saucepan, and cook over medium heat, stirring constantly, until it returns to a boil, 1 to 2 minutes.
Remove from heat, and stir in 1/4 cup rum. Add butter, 1 tablespoon at a time, whisking until butter melts before adding next piece. Let cool in saucepan on a wire rack, whisking occasionally, for about 10 minutes.
Pour custard into piecrust. Press plastic wrap directly on surface of custard. Refrigerate until custard filling is chilled and firm, at least 4 hours (or overnight).
Meanwhile, place cream and vanilla pod in a bowl. Cover, and refrigerate for at least 4 hours (or overnight).
Just before serving, remove vanilla pod from cream, and discard. Whisk cream until soft peaks form. Add confectioners' sugar and remaining tablespoon rum, and whisk until stiff peaks form. Transfer to a pastry bag fitted with a 1/2-inch star tip (such as Ateco #826), and pipe whipped cream over pie. Serve immediately.