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In a medium bowl, whisk together the flour, baking powder, and salt. Set aside.
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In a large bowl, use your fingers to rub the sugar and lemon zest together until the sugar is moist and fragrant.
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Add the eggs to the sugar mixture and beat on medium high speed until pale, thick, and light (2-3 minutes). Beat in the vanilla extract.
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Using a rubber spatula, gently fold the flour mixture into the egg mixture. Once combined, fold in the butter.
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Press a piece of plastic wrap directly onto the surface of the batter and refrigerate for at least 3 hours and up to 2 days.
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Preheat the oven to 400 F. Grease the madeleine pan and spoon the batter into the pan, filling each one almost to the top.
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Bake for 10-12 minutes or until golden and the tops spring back when touched.
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Remove the pan from the oven and gently tap over a wire rack to release the madeleines.
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Let cool until just warm or room temperature. Just before serving, dust madeleines with powdered sugar. Madeleines are best eaten the same day they are made.