Add the eggs and sugar to the bowl of a stand mixer fitted with the paddle attachment. Beat it on medium-high speed for 8-10 minutes, or until it is smooth, pale yellow, and creamy. Add vanilla the last minute of beating.
In a separate bowl, combine the flour, cocoa powder, and baking powder.
Combine the flour mixture with the egg mixture and fold it in gently.
Melt the butter - about 2 minutes in the microwave - and fold it into the batter.
Transfer the batter in the fridge for 1 hour (I don’t usually cover mine but use plastic wrap or a kitchen towel if you prefer to cover it).
Towards the end of the chilling step, preheat the oven to 350° F and grease a madeleine pan with melted butter and non-stick cooking spray.
Scoop the batter into the molds until they are ¾ full using a spoon or ice cream scoop. Gently press the batter into the molds (wetting your hand helps prevent the batter from sticking).
Bake until the madeleines are set - about 13 minutes.
Transfer the madeleines to a wire rack to cool.
If desired, garnish with a light dusting of powdered sugar or dip in melted chocolate.
Enjoy immediately or store in an airtight container at room temperature (or in the fridge if you dipped them in chocolate) for up to 3 days.