Beat together the eggs, sugar, zest, and vanilla in a large bowl for several until pale and fluffy.
Sift the flour, baking powder, and salt into the bowl and fold in with a rubber spatula until you have a smooth batter.
Remove some of the batter and mix it into the melted butter, then slowly add the melted butter and batter mixture back into the rest of the batter and mix gently until combined.
Cover and refrigerate the batter for 30 minutes or freeze for 15 minutes. You can refrigerate the batter for up to 24 hours before baking but do not freeze for longer than 15 minutes. Do not skip chilling the batter, this is what helps them rise.
Preheat the oven to 400°F and lightly grease the madeleine pan. Since the pan has scalloped edges, the best way to do this is with a pastry brush and melted butter, but you can also use cooking spray. There's no need to flour the pan, in fact, they taste better without it in my opinion.
Take the batter out of the fridge and quickly mix again to remove excess air bubbles.
Fill your madeleine form evenly with the batter. Don´t fill them too high, the dough will rise a lot. For me, a full tablespoon works perfectly.
Bake on the middle rack for 10 to 12 minutes or until the centers bump up and they are golden brown around the edges. The madeleines should have a little bump on the top side, this is the desired "pearl" and what makes them unique.
Let them cool in the pan for a few minutes then transfer them to a wire rack to cool completely. Dust with powdered sugar before serving, if desired.