Place 3 whole eggs plus 3 egg yolks, vanilla extract, sugar, and rum into a bowl and beat until combined.
In a pot, heat up the milk. When the milk is hot, slowly transfer into the egg, sugar, vanilla extract and rum mixture while whisking. If you have a stand mixture, leave the mixture running while adding the hot milk. Don't add the hot milk too quickly into the egg mixture, as it will scramble the eggs.
After the milk and egg mixture is combined, strain it then pour on top of the caramel sauce.
Place a big aluminum pan in a preheated oven first, and gently put the cake pan containing the custard inside the aluminum pan. Then fill the aluminum pan with hot water. Bake for 45 minutes or until an inserted toothpick comes out clean.
After the flan is baked, remove it from the water bath to cool. Before serving, run a knife around the edge of the cake pan to loosen the flan.
Place a plate on top of the flan and quickly turn it over. The flan should come out easily. Enjoy.