Combine sugar, cornstarch, and salt in a 2-quart saucepan. Stir in milk gradually. Add chocolate chips and unsweetened chocolate. Place over medium heat, stirring constantly, until mixture thickens and boils. Boil and stir 1 minute.
Place egg yolks in a medium heatproof bowl. Gradually pour half of chocolate mixture into egg yolks, whisking constantly.
Whisk egg yolk mixture back into mixture in saucepan. Place over medium heat and bring back to a boil, stirring constantly. Boil and stir 1 minute. Remove from heat; stir in vanilla extract.
Pour mixture into baked pie shell. Press a layer of plastic wrap onto filling. Refrigerate at least 4 hours but no longer than 48 hours. Remove plastic wrap before serving and top with whipped topping.