Whisk together the eggs, caster sugar, flour, milk, rum or vanilla extract and vegetable oil until smooth and creamy. Place batter into the refrigerator to rest for at least 1 hour.
Heat a frying pan with a small amount of butter over medium heat, and pour in a ladle of the batter. Quickly tilt the frying pan to allow the batter to cover the surface completely. Cook until golden, about 2 minutes, then turn and cook on the other side until golden, about 2 minutes more.