Combine the cocoa powder, cornstarch, sugar, and salt in a large saucepan over medium heat. Slowly whisk in the half-and-half, then beat in the egg yolks and whisk in the mild.
Let the mixture come to a simmer, whisking constantly. Let the mixture simmer until it starts to thicken, about 2 minutes. Add the chocolate and stir until it melts.
Reduce the heat to medium and continue to cook whisking frequently, until the pudding is fairly thick, about 2 minutes. It will thicken more upon cooling.
Whisk in the butter and vanilla, until the butter is melted. Transfer to a serving bowl or individual serving dishes.