In a heavy medium saucepan, bring 2 cups of milk to simmer.
Remove from heat.
Whisk egg yolks, brown sugar, vanilla and remaining ¼ cup milk to blend in a medium bowl.
Whisk in cocoa powder, instant espresso, if using, and cornstarch until smooth.
Gradually mix in hot milk mixture, return entire mixture to sauce pan, whisk constantly over medium heat until mixture thickens and boils.
Remove from heat, add chopped chocolate and butter and whisk until melted and smooth.
Pour filling into a blind baked and cooled pie crust.
Refrigerate until filling is cold and set, about 4 hours.
Chef's notes: Can be prepared a day ahead.