Whisk eggs and sugar together in a bowl.
Sift in cake flour and baking powder, then mix.
Add melted butter and vanilla extract, and combine well.
Rest the batter in the refrigerator for at least 30 minutes.
Brush the madeleine pan with butter and fill with the batter.
Bake in a preheated oven at 180°C (356°F) for 10-12 minutes.
Cool on a wire rack.