In a medium saucepan, melt the unsalted butter over medium heat. Continue to cook, stirring frequently, until the butter turns golden brown and has a nutty aroma. Remove from heat and let cool slightly.
In a mixing bowl, whisk together the all-purpose flour, baking powder, and salt. In another bowl, whisk the room temperature eggs and granulated sugar until light and frothy. Add the lemon zest and vanilla extract, mixing until well combined.
Gradually add the flour mixture to the egg mixture, folding gently with a spatula. Finally, add the browned butter, folding until just combined. Be careful not to overmix.
Cover the bowl with plastic wrap and refrigerate the batter for at least 30 minutes. This helps the Madeleines achieve their characteristic rise and texture.
Preheat your oven to 375°F (190°C). Prepare your Madeleine pan by brushing it with melted butter to ensure easy release.
Spoon the chilled batter into each shell of the prepared pan, filling each cavity about 3/4 full. Bake for 10-12 minutes or until the edges are golden and the centers spring back when lightly touched.
Remove the Madeleines from the oven and let cool in the pan for a few minutes before transferring them to a cooling rack. Dust with powdered sugar before serving. Enjoy with tea or coffee!