In a large saucepan, whisk whole milk, sugar, heavy cream, large egg, large egg yolk, cornstarch and natural unsweetened cocoa powder. Cook over medium-high, whisking constantly, until the mixture thickens and begins to bubble around the edges of the pan, 9 to 11 minutes. Remove from the heat; whisk in chopped bittersweet chocolate (60 to 72% cacao), butter and pure vanilla extract. Using a rubber spatula and working in batches if necessary, press the pudding through a fine-mesh strainer into a medium bowl. Let cool 10 minutes, then press plastic wrap directly on the surface of the pudding to prevent a skin from forming. Refrigerate until cold, about 2 hours.