Grease a 12-cavity madeleine pan and place in the fridge.
Place the butter in a small saucepan. Melt over medium-low heat and bring to a simmer. Continue to cook until the butter turns golden brown, with a nutty aroma. Remove pan from heat and set aside to cool slightly.
Beat the eggs till just blended, then add the sugar and vanilla. Beat on high for 5 minutes. Slowly add the brown butter (being sure to scrape up any of the solids at the bottom of the pan). Add the flour, baking powder, baking soda and salt and beat till blended. Chill the batter for 1 hour.
Pre-heat the oven to 375 degrees.
Place the batter in a disposable pastry bag and snip an opening in the corner. Pipe the batter evenly into the wells in the pan. Bake for 8 - 12 minutes or till golden brown around the edges. Cool the madeleines in the pan on a wire rack for 5 minutes, then remove to rack to cool completely.
For dipped and decorated madeleines I suggest a silicone pan. Melt the chocolate in a bowl set over a pan of barely simmering water and transfer to a disposable pastry bag. Place a flower in each of the wells and pipe the melted chocolate over to cover. Gently press the cooled madeleine into the chocolate and let set completely before removing.
Notes
A metal pan is generally preferred for baked goods with crisp-tender edges. But if you're going to dip the madeleines you'll have a much easier time using a silicone pan!