Preheat your oven to 375°F (190°C). Grease your madeleine molds with butter or cooking spray, ensuring every crevice is well coated to prevent sticking.
In a mixing bowl, whisk together the flour, baking powder, and salt. Set aside.
In a separate bowl, combine the sugar and eggs. Using an electric mixer, beat the mixture on medium-high speed until it becomes pale and fluffy, which should take about 2-3 minutes. The mixture should leave a ribbon-like trail when the beaters are lifted.
Add the vanilla extract to the egg mixture and gently fold it in.
Gradually add the flour mixture to the egg mixture, folding gently until just combined. Avoid overmixing, as this can result in dense madeleines.
Slowly pour the melted butter into the batter, stirring gently until fully incorporated. The batter should be smooth and shiny.
Cover the bowl with plastic wrap and refrigerate for at least 1 hour, or up to overnight. Chilling the batter helps the madeleines develop their characteristic humps.
Once the batter has chilled, fill each madeleine mold with about 1 tablespoon of batter. Don't worry about spreading the batter evenly – it will naturally fill the molds as it bakes.
Bake the madeleines for 10-12 minutes, or until the edges are golden brown and the centers are set. Avoid overbaking, as this can dry out the madeleines.
Remove the madeleines from the oven and let them cool in the molds for a few minutes. Gently tap the molds to release the madeleines, and transfer them to a wire rack to cool completely.