Preheat oven to 350°F. Line a baking sheet for the springform pan to sit on (optional for easier handling).
In a medium bowl, stir together 2 cups chocolate wafer crumbs and 2 1/2 tablespoons sugar. Stir in 1/3 cup melted butter until evenly moistened.
Transfer the crumb mixture to a 9-inch springform pan. Press the crumbs firmly and evenly onto the bottom and up the sides about 1 1/2 inches.
Place the springform pan on the prepared baking sheet and bake the crust at 350°F for 10 minutes. Remove from the oven and let the crust cool while you prepare the filling.
Put 2 cups (12 ounces) semisweet chocolate chips in a microwave-safe bowl. Heat in 20-second intervals, stirring after each interval, until the chocolate is completely smooth. Let the melted chocolate cool for a few minutes.
In a medium bowl, whisk together 6 egg yolks, 1 3/4 cups heavy whipping cream, and 1 teaspoon vanilla extract until combined.
With the egg mixture in a steady whisking motion, gradually pour the slightly cooled melted chocolate into the egg mixture to combine (this tempers the eggs). Whisk until the mixture is smooth and uniform.
Pour the chocolate custard into the prepared crust. Bake at 350°F for 45 to 50 minutes, or until the center is almost set (it may still have a slight jiggle).
Remove the pan from the oven and let the dessert cool on a wire rack for 10 minutes. Run a knife around the inside edge of the springform pan to loosen the dessert.
Let the dessert cool to room temperature, then cover and refrigerate overnight.
To make the buttercream, beat 1/2 cup softened butter in a bowl with an electric mixer until smooth.
With the mixer on low, gradually add 3 cups confectioners’ sugar and 3 tablespoons unsweetened cocoa powder until mostly incorporated.
Add 3 tablespoons milk and beat until smooth. If the frosting is too thick to pipe easily, add up to 1 additional tablespoon milk (up to 4 tablespoons total) until you reach a piping consistency that holds its shape but can be squeezed from a piping bag.
If desired, release the springform pan sides and transfer the chilled dessert to a serving plate. Fit a pastry bag with a large star tip, fill with the buttercream, and pipe stars all over the top of the chilled dessert.