Beat butter and icing sugar till smooth. Add in the egg yolk and egg white and mix.
Add in vanilla essence, then the flour, and mix into a soft dough. Wrap the dough in plastic and chill it in the refrigerator for 15 minutes.
Press dough into small tart moulds or muffin pans.
For egg custard, heat sugar and milk together over a small flame. Once the sugar dissolves, turn off the heat. Cool to room temperature.
Lightly whisk the eggs and pour into the milk mixture, whisking a little. Do not whisk till frothy.
Stir in the rum and vanilla essence.
Pass the egg custard through a sieve and pour into the tart moulds or muffin pans and bake on the lower shelf of your oven at 175 degrees C for 20 minutes. Rotate the pan and bake for 10 more minutes until egg custard sets. You will know that your custard has set when it does not wobble when you shake the pan. If you see that the custard is about to puff up into a ball, take it out from the oven immediately. If you allow the custard to puff up, the egg tart will wrinkle when it's cooled.
The egg tarts may be left at room temperature for one to two days. Try not to refrigerate them, as the crust will harden.