Melt butter in a skillet on medium heat, stirring periodically. First you'll see large bubbles, then you will see smaller bubbles, and then it will start to brown. Remove from the heat as soon as it starts to brown and let it cool to room temperature while you prepare the batter.
Preheat oven to 375 F.
In a small mixing bowl, beat eggs, vanilla extract, and salt until fully combined.
Beating constantly, gradually add sugar and brown sugar. Continue beating for ten minutes at high speed until mixture is thick and pale and ribbons form in the bowl when the blade is lifted.
Add cake flour into egg mixture 1/3 at a time, gently folding by hand after each addition.
Pour melted browned butter around the edge of the batter and quickly but gently fold into the batter,
Fill Silpat's mini madeleine molds 3/4 full using two spoons - one to get a spoonful of batter from the bowl, and one to scrape the batter off of the spoon and into a mini madeleine well.
Bake 11 minutes, or until cakes are golden and the tops spring back when gently pressed with your fingertip. Then, set on a cooling rack to cool.
Serve the madeleines warm and dusted in powdered sugar, dip them in melted chocolate, or use them to make a madeleine tower.
If you are making full-sized madeleines rather than minis, this recipe will make about 24 madeleines.If you are not using a Silpat mold, be sure to both butter and flour the madeleine wells before baking. The madeleines will be difficult to remove if you don't grease the wells properly.Store in an airtight container. They are best served on the same day, but they will still be great (especially dunked in coffee) for several days.Try adding a little lemon zest, orange zest, matcha powder, or spices to your madeleines for other flavor options.