Combine the milk, cream and cinnamon sticks in a saucepan and bring to the boil. Remove from the heat.
Whisk the egg yolks, vanilla sugar and sugar until it has thickened, then add the milk.
Stir until thoroughly combined. Add the rum and refrigerate overnight (keeps for up to 3 days).
Before serving, whisk the egg whites into a foam and fold into the egg toddy. Serve cold.