Add butter in a nonstick saucepan and melt it over medium heat. You can choose to brown the butter and that will add a lovely nutty flavor and aroma. Remove from heat and keep aside to cool as you work on the remaining steps.
In a large mixing bowl, add eggs, sugar, lemon zest, vanilla extract and mix for 5-6 minutes until mixture becomes thick and pale and it forms ribbons when you lift the beaters.
Add half the baking powder and half the flour and gently mix it into the wet mixture with a silicon spatula. Do not use the mixer any more!
Add the remaining baking powder and remaining flour and gently gently mix it into the wet mixture with a spatula too until just combined.
Add melted butter and gently stir it into the batter too until mixture is smooth and shiny.
Cover and chill in the fridge for 40 minutes. This allows it to become airy and "spongey."
Brush melted butter lightly on your madeleine pan.
Remove batter from fridge. Do not mix! That will cause air bubbles to deflate.
Spoon out about 1 tablespoon of the batter and drop it in the middle of each mold. Don't spread it.
Bake at 350 degrees F for 10-12 minutes or until tops are light golden brown and there is a "hump" or "bump" or "belly" in the center. They should be slightly firm along the edges and an inserted toothpick in the center should come out clean.
Remove from oven and let them cool in the pan for about 1 minute and then, gently remove them from the molds and transfer them to a wire rack to slightly cool.
Top them off with a light dusting of powdered sugar.
Serve immediately while warm and enjoy.