Heat oven to 350°. Place pie crust in the bottom of a 9-inch pie plate. Bake for 20 minutes; let cool completely.
Melt the chocolate in a small bowl in the microwave in 30-second intervals. When the crust is cool, pour the chocolate over it and spread it out to cover the entire bottom. Allow the chocolate to cool completely.
In a saucepan, whisk together sugar, cornstarch, and salt. Add egg yolks, milk, and cream and whisk until combined.
Set the saucepan over medium heat on the stove. Whisk occasionally until the mixture begins to thicken, about 3 minutes. Cook 2-3 minutes more, whisking constantly, until bubbles start to form and the mixture is very thick.
Remove pan from heat and whisk in vanilla. Add butter and chopped chocolate in small batches, whisking until melted and smooth.
Place a sieve over a medium bowl and strain the custard, pressing it through with a rubber spatula. Cover with plastic wrap, pressing plastic onto the surface; refrigerate until set, about 3 to 4 hours.
Remove plastic wrap from custard and stir until smooth. Spoon mixture into cooled crust and smooth surface with a spatula.