Combine Dry Ingredients. In a medium saucepan, combine cornstarch, cocoa powder, sugar, and salt. Whisk to combine well.
Add Milk & Cook To Thicken. Next, pour in the milk and whisk to combine, stirring over medium heat until the mixture comes to a simmer. At first, it will seem like the cocoa powder mixture isn't going to blend in, but as the milk heats up, it will dissolve the cocoa until the mixture is smooth. Whisk constantly to avoid lumps. It's VERY important to get the filling to thicken enough that you can see the tracks the whisk makes in the pudding. If you don't thicken the pudding at this stage, it will not set up later.
Finish The Pudding. Remove the pan from the heat and immediately stir in the butter, chocolate, and vanilla. Whisk to combine until the mixture is completely smooth and the chocolate has melted. (If you do have any lumps in the chocolate mixture at this point, you can press the pudding through a fine mesh strainer to strain them out.)
Pour Filling Into Crust. Working quickly (so the pudding doesn't form a "skin" on the top), pour the warm pudding into the cooled baked crust. Smooth out the top with a spatula.
Cover And Chill. Cover the pie with plastic wrap, pressing the plastic wrap down onto the pudding (Again, this helps prevent a skin from forming on the top of the pie). Chill at least 2-3 hours, or until chilled through. (You can prep this pie up to a day in advance, if needed.)