In a mixing bowl, whisk together the sugar, cocoa, cornstarch, and salt. In a thin stream, pour in the milk whisking until smooth. In a thin stream, add in the half-and-half and whisk until smooth.
Pour the mixture through a fine sieve or chinois into a saucepan. Over medium heat, whisking the mixture constantly, bring to a boil. Boil gently 2 minutes. Turn off the heat and stir in the bittersweet chocolate and vanilla extract.
Transfer the custard to a mixing bowl.
Fill a large bowl with ice cubes, rest the mixing bowl on top, and add cold water to cover the ice cubes.
Let the mixture cool, stirring frequently with a rubber spatula, about 15 minutes.
Fold in the white chocolate and pour the mixture into 1-cup ramekins, cups or mugs.
