In a medium saucepan, whisk together the sugar, cocoa powder, cornstarch, and salt until no lumps remain. This prevents clumping when you add the liquid.

Whisk in about 1 cup of the milk to form a smooth paste, then whisk in the remaining milk, cream, and egg yolks until fully combined. Scrape the corners of the pan where dry ingredients hide.

Set the pan over medium heat and cook, whisking constantly and scraping the bottom and sides, until the mixture thickens and large bubbles begin to pop on the surface. This takes 6 to 8 minutes. The pudding should coat the back of a spoon and hold a line when you run your finger through it.

Remove from heat and immediately whisk in the butter, vanilla, and chopped chocolate until completely melted and the pudding is glossy and smooth.

Pour through a fine-mesh sieve into a clean bowl or individual ramekins to remove any cooked egg bits. Press plastic wrap directly onto the surface to prevent a skin from forming. Refrigerate until cold and set, at least 2 hours.

Serve cold with lightly sweetened whipped cream and chocolate shavings if desired. The pudding keeps well for up to 4 days.