1. Whisk Dry Ingredients. In a medium saucepan, combine sugar, cocoa powder, cornstarch, and salt. Whisk to combine well. 
2. Pour In Milk. Add the milk and whisk to combine. It will take a second to combine, and it will look pretty foamy at first. 
3. Cook To Thicken. Whisk the pudding over medium heat until the mixture begins to thicken. You want to stir pretty constantly, so the pudding doesn’t form lumps. You’ll know it’s thickened when any foamy bubbles go away, and you can see the path of the whisk in the pudding as you stir. Remove from the heat. 
4. Finish. Immediately add chocolate, butter, and vanilla extract. Whisk to combine until the chocolate is melted. If desired, you can strain the pudding through a fine mesh sieve to remove any small lumps, but I don’t usually. 
5. Cover & Chill. Transfer the pudding to a large bowl or individual serving cups. Cover with plastic wrap or food wrap so the pudding doesn’t develop a “pudding skin” (a layer of thickened pudding on top). Chill in the refrigerator 2-3 hours or until completely cool. 
6. Serve & Store. Enjoy the pudding cold, or serve with a dollop of whipped cream and shaved chocolate for garnish. Leftover chocolate pudding will keep covered in the fridge 3-4 days. 

The pudding will continue to thicken as it chills. If you want to loosen it up a bit before serving, feel free to whisk it until it loosens up. (I almost never do this unless I chilled it in one large bowl and want to spoon it into smaller dishes for serving.)