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Butter and flour a madeleine pan with 12 cavities.
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In a small bowl, combine the flour and baking powder. Set aside.
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In the bowl of an electric mixer, whisk the eggs by hand. Whisk in the sugar in a stream. Whisk in the lemon zest and vanilla.
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Place the bowl on the mixer fitted with the whisk attachment and whip the mixture on medium-high speed until it is light and fluffy, about 2 to 3 minutes.
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Remove the bowl from the mixer and fold in the flour mixture. After the flour has been absorbed, fold in the butter.
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Cover the bowl and chill 3 hours or overnight (this helps them develop the characteristic humps on their undersides, but you can proceed immediately to the baking if you don’t mind them being flat).
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When you are ready to bake, preheat the oven to 400 degrees.
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Use a spoon to fill each cavity of the pan about ¾ full.
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Bake the madeleines until they are light golden and feel firm when pressed with a fingertip, about 10 to 12 minutes.
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Invert the pan onto a work surface and gently bang one of the sides of the pan to help the madeleines fall out, then transfer them to a rack, grooved side up, to cool completely.