Martha prefers to eat this classic tart long after dinner is done, when it can be truly enjoyed; its buttery crust, chocolate pudding, and meringue-like filling deserve their own spotlight, almost like a second meal.
Preheat oven to 400 degrees. On a lightly floured surface, roll out pate brisee disk into a 13-inch round about 1/8 inch thick. Fit dough into a 9-inch bottomless tart ring set on a baking sheet. Using a sharp paring knife, trim the dough flush with the rim. Prick the bottom of the dough all over with a fork. Transfer to the freezer or refrigerator, and chill until firm, about 30 minutes.
Remove baking sheet from freezer. Line tart ring with parchment paper, leaving about 3 inches overhang. Fill with pie weights or dried beans. Bake 20 minutes. Remove from oven, and remove weights and parchment paper. Return to oven; continue baking until crust is deep golden, 10 to 15 minutes. Transfer to a wire rack to cool completely.
In a microwave or a medium heatproof bowl set over a pan of simmering water, melt the chocolate; set aside. In a small bowl, sprinkle gelatin over the cold water; let stand until softened, about 5 minutes.
In a medium bowl, whisk together egg yolks, 1/4 cup sugar, and cornstarch; set aside. In a medium saucepan, combine milk and 1/4 cup sugar; cook over medium heat until mixture begins to bubble around the edge, about 3 minutes. Whisking constantly, gradually add hot milk mixture to egg mixture. Return mixture to pan; cook, stirring constantly over medium to medium-low heat until thick and just starting to boil, 5 to 7 minutes.
Prepare an ice bath. Measure out 1/2 cup custard mixture, and place in a medium bowl; immediately whisk in gelatin mixture until smooth. Place bowl in the ice bath several seconds to let mixture set. Remove from ice bath; place a piece of plastic wrap directly on surface to prevent a skin from forming. Set aside.
Stir remaining custard into melted chocolate, and add vanilla. Set bowl in ice bath; let cool completely, stirring occasionally. Spread mixture evenly in bottom of prepared tart shell. Refrigerate until set, at least 5 minutes.
In the bowl of an electric mixer fitted with the whisk attachment, beat egg whites with salt and cream of tartar on medium speed until soft peaks form, about 2 minutes. With mixer running, gradually add remaining 1/4 cup sugar; beat until peaks are stiff and glossy, 1 to 3 minutes. Whisk one-third meringue into reserved custard mixture until completely smooth. (You may need to beat the custard again at this point until smooth.) Gently but thoroughly fold remaining meringue into custard mixture. Spread over chocolate layer in tart shell. Cover with plastic wrap; refrigerate at least 3 hours or overnight.
In a small bowl, whip heavy cream until stiff peaks form. Transfer to a pastry bag fitted with a star tip (Ateco #825), and pipe decoratively over custard (alternatively, spread with an offset spatula). Sprinkle chocolate shavings on top. Refrigerate until ready to serve.