Preheat oven to 350 degrees. Line dough with parchment, and fill with pie weights or dried beans. Bake until edges begin to look dry, 20 to 22 minutes. Carefully remove weights and parchment. Bake until crust is darker around edges and bottom looks dry, 10 to 12 minutes. Let cool completely on a wire rack. (Chocolate crust can be stored overnight.)

Whisk together sugar, cornstarch, and salt in a small bowl. Heat milk and chocolate in a medium saucepan over medium-high heat, stirring occasionally, until chocolate melts.

Whisk 1 cup hot milk mixture into sugar mixture until smooth. Whisk milk-sugar mixture into the remaining milk mixture in the saucepan. Cook over medium heat, stirring constantly, until bubbling and thick, 4 to 5 minutes total (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 mixture to saucepan. Cook over medium heat, stirring constantly, until it just begins to bubble, 1 to 2 minutes.

Remove from heat, and stir in vanilla. Let custard cool in saucepan on a wire rack, whisking occasionally, for about 10 minutes.

Pour custard into chocolate crust. Press plastic wrap directly on surface of custard. Refrigerate until custard filling is chilled and firm, at least 4 hours (or overnight).