Sift cocoa powder into a medium saucepan and blend in the sugar. Gradually whisk in 1½ cups of the milk until fully combined and smooth. Place the saucepan over medium-high heat and stir continuously until the mixture reaches a gentle simmer, then remove from heat. In a separate large bowl, vigorously whisk the remaining ½ cup milk with the cornstarch, egg yolks, and salt until silky. While whisking steadily, slowly pour the hot cocoa mixture into the egg yolk blend to temper. Return the combined mixture to the saucepan over medium-high heat and whisk without pause until it reaches a rolling boil. Immediately decrease the heat to maintain a gentle simmer and continue whisking until the pudding thickens, about 2 to 3 minutes longer. Remove from heat and incorporate the melted chocolate chips, softened butter, and vanilla, stirring thoroughly until glossy and uniform. Pass the pudding through a fine mesh sieve to ensure a smooth finish, then distribute evenly into four (4-ounce) cups or bowls. Press plastic wrap directly onto the surface of each pudding to prevent a skin from forming. Refrigerate for a minimum of 4 hours or overnight until set. Before serving, crown each pudding with freshly whipped cream and chocolate shavings, if desired. Store any remaining pudding in the refrigerator, tightly covered, for up to four days.