1. To make the crust: Lightly butter a 10-inch pie plate. On a lightly floured board, roll out the half of the pate brisee to a 1/8-inch thickness (unused dough can be wrapped and plastic and stored in the refrigerator or freezer). Place the dough in the pie plate, and press it into the bottom edges and along the sides. Trim the dough using scissors or a sharp paring knife. Crimp or decorate the edges of the crust if desired. Chill crust for 30 minutes.

2. Preheat the oven to 375 degrees. Carefully line the pie plate with parchment paper, gently pressing the parchment into the edges of the crust; weigh down parchment with dried beans or pie weights. Bake until edges of pastry begin to turn golden, about 15 minutes. Carefully remove pie weights and 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.

3. To make the filling: Prepare an ice-water bath. In the top of a double boiler or in a heat-proof bowl set over simmering water, melt chocolate; set aside. Place 1/4 cup cold water in a small bowl, and sprinkle gelatin over it; set aside to soften.

4. In a medium bowl, whisk together 4 egg yolks, 1/4 cup sugar, and the cornstarch; set aside. In a medium saucepan, combine milk and 1/4 cup sugar. Bring just to a boil. Add about one-quarter of the hot-milk mixture to the egg mixture, whisking constantly, until combined. Return egg-milk mixture to saucepan, and bring to a boil. Remove from heat when mixture has boiled and thickened, 3 to 4 minutes.

5. Measure out 1 cup of custard, and transfer to a medium bowl. Add the melted chocolate and vanilla to it, stirring until the chocolate is melted. Transfer the bowl to the ice-water bath to cool completely. When cool, spread evenly in the bottom of the pie crust. Transfer pie crust to refrigerator until set, about 5 minutes. Add softened gelatin to remaining custard. Transfer pan to the ice-water bath to cool until just beginning to set.

6. In the bowl of an electric mixer fitted with the whisk attachment, beat egg whites with salt and cream of tartar until soft peaks form. With machine running, gradually add remaining 1/2 cup sugar, and beat until stiff glossy peaks form. Stir 1/3 of whipped egg whites into custard to lighten. Gently but thoroughly fold in remaining whites and rum. Spread over chocolate layer. Refrigerate pie for at least 3 hours and up to overnight.

7. Whip cream until stiff peaks form. Spread over custard or pipe on top with a pastry bag fitted with a star tip. Sprinkle chocolate shavings on top. Chill until ready to serve.