Add milk into the sugar mixture. Stir until a paste. Combine milk and egg yolks.
Cook the mixture on a low heat, while stirring continuously. Let the mixture thicken and don't over boil. Turn off the heat. Add butter, rum and vanilla.
Transfer mixture to a piping bag fitted with a 230 piping tip.
Pipe pudding into each doughnut Hole. (You might have to make some room inside the doughnut hole with the tip in order to fill with the pudding.)