First, melt the butter and allow it to cool while you make the batter.
In a small bowl place the flour, baking powder and salt and whisk until well blended.
In the bowl of your electric mixer, beat the eggs and sugar at medium-high speed until the mixture has tripled in volume and forms a thick ribbon when the beaters are lifted (about 5 minutes).
Add the vanilla extract and beat to combine.
Sift a small amount of flour over the egg mixture and, using a large rubber spatula, fold the flour mixture into the beaten eggs to lighten it. Sift the rest of the flour over the egg mixture and fold in being sure not to overmix or the batter will deflate.
Whisk a small amount of the egg mixture into the melted butter to lighten it. Then fold in the cooled melted butter in three additions. Cover and refrigerate for at least 30 minutes, or several hours, until slightly firm.
Position a rack in the center of the oven and preheat to 375 degrees F (190 degrees C). Generously butter two 12-mold madeleine pans. Dust the molds with flour and tap out the excess. (Could spray pans with Bakers Joy instead.) (Make sure the pans are well greased or the madeleines will stick and be hard to remove.)
Drop a generous tablespoonful of the batter into the center of each prepared mold, leaving the batter mounded in the center. (This will result in the classic "humped" appearance of the madeleines.)
Bake the madeleines for 11 to 13 minutes, until the edges are golden brown and the centers spring back when lightly touched. Do not overbake these cookies or they will be dry.
Remove the pans from the oven and rap each pan sharply against a countertop to release the madeleines. Transfer the madeleines, smooth sides down, to wire racks to cool. The madeleines are best served the same day but can be stored in an airtight container at room temperature for 2 to 3 days or frozen, well wrapped, for up to 1 month.