In a medium bowl, whisk egg yolks for a minute or so. They'll turn a lighter yellow color and increase in volume. Then, whisk in cornstarch and ¼ cup milk. Whisk until smooth and set aside.
Place sugar, cocoa, salt, and remaining milk in a saucepan and bring to a scald over medium-high heat. Bubbles will form around the sides of the pan but does not reach a full boil (about 180°F). Remove from heat.
Slowly temper a ladle of hot cocoa mix into egg mixture, whisking constantly. Repeat with another ladle while whisking and slowly tempering.
Pour everything back into the saucepan and cook over medium-high heat, whisking constantly, until the pudding comes to a full boil. Reduce heat to a simmer and continue whisking for a couple minutes until the pudding is thickened.
Remove pudding from heat and stir in the chocolate and vanilla extract. Pour pudding into a bowl and press plastic wrap directly on the surface of the pudding so a skin doesn't form. Refrigerate for an hour or two until chilled.