Melt the butter in a small sauce pan on medium heat until it's brown and starts to smell rather nutty. Do make sure that you don't let it get burned as that.... is something you do not want! Once brown, strain the butter (you can use a paper towel over a mesh strainer) so that the solids are left behind. Cool the butter to room temperature. By doing the butter first you can complete the rest of the steps while it is cooling.
While the melted butter is cooling, use the remaining 2 tablespoons of butter to grease the madeleine molds - get in there and make sure you get in all the ridges. Dust with flour and invert the pan tapping out any excess flour. Mine came out of the pan very easily.
Put the eggs with the salt in the bowl of an electric mixer with a whisk attachment. Whip on high speed until thick - you are looking for the eggs to roughly double or triple in volume - approximately 3 minutes. Continuing to mix on high speed, slowly add the sugar in a steady stream. Whip for 2 minutes or until mixture is thick and ribbony. Now with a spatula fold in the lemon zest and vanilla (just until mixed).
Sprinkle the flour on top of the egg batter, and gently fold in. Now fold in the butter mixture. Only stirring enough to bring everything together. You don't want the butter to be too hot or it will work on your eggs.
Spoon the batter into the molds, filling each mold 2/3 -3/4 full. I found that the batter becomes thick quite quickly. I had to bake in several batches as the mold only holds 12 madeleines. This recipe makes double that amount.
Bake the madeleines for about 10-15 minutes depending on your oven. Make sure to check them from time to time. Once the edges turn golden remove from the oven and turn out onto a cooling rack immediately. Dust with icing sugar and serve. The madeleines are best eaten the day they are made.