In a medium saucepan set over medium heat, whisk together granulated sugar, cocoa powder, cornstarch, salt, and optional instant coffee granules until well combined.

Gradually pour in the milk, whisking continuously to create a smooth, lump-free mixture.

Increase the heat slightly and bring the pudding mixture to a gentle simmer, stirring constantly.

Once the pudding has thickened, remove it from heat and immediately stir in the butter and vanilla extract.

Pour the finished chocolate pudding into a storage container, optionally using individual cups and press plastic wrap onto the surface.

Place the pudding in the refrigerator and let it chill for at least 2 hours before serving.