In a large saucepan, whisk the sugar, cornstarch, cocoa, and salt together.
⅓ cup granulated sugar, ¼ cup cornstarch, 2 tablespoons unsweetened cocoa powder, ½ teaspoon kosher salt
Add the milk and whisk, scraping the sides until fully incorporated.
3 cups whole milk
Set the saucepan over medium to medium-low heat. Cook and whisk continuously for 8-10 minutes, or until bubbly over the entire surface and thickened.
Cook an additional 30 seconds and remove from the heat.
Add the chocolate and butter and whisk until melted and smooth. Add the vanilla and whisk until incorporated.
4 ounces semisweet chocolate, 2 ounces bittersweet chocolate, 3 tablespoons unsalted butter, 2 teaspoons pure vanilla extract
Pour the filling into the cooled pie crust and cover with the prepared parchment. Lightly press the parchment against the surface of the filling.
Cool the pie on the countertop 45-60 minutes then refrigerate until filling is set and firm, about 3 hours. The pie can be made up to 24 hours ahead, but add the whipped cream not more than 2 hours before serving.