Melt the butter and set it aside to cool for at least a half hour.
In a bowl, sift together the flour, cocoa powder, baking powder.
In a stand mixer with the whisk attachment or with electric beaters, beat the eggs with the sugar until they are very pale yellow. This will take 4 or 5 minutes.
Fold in the flour mixture. When that is nearly all mixed in, pour in the butter and fold that in until it is fully incorporated.
Put the batter in the fridge for about an hour to chill.
Preheat the oven to 375 F.
Scoop in approximately 2 tbsp of batter into each well in a madeleine pan. Do not spread the batter, it will melt in the oven. Put the bowl back in the fridge if you don't have enough pans to bake all the batter in one go.
Bake at 375 F for 10 minutes or until the madeleines are springy to the touch in the centre. If they are still soft and your finger leaves a dent then put them back in the oven for another minute.
Remove them from the pan and cool them on a rack.
Cool the pan with cold water before filling the pan again.