Let the butter melt, bubble and foam until it begins to brown (beurre noisette) and gives off a nutty caramel-like aroma (about 5 minutes). You will need to swirl the pan every now and then to ensure even cooking. Do not worry when you see specks of dark brown and black in the butter- the milk solids tend to cook faster than the fat.
Let the butter cool completely.
In the meantime, stir together the flour, baking powder and salt, then set aside.
In a bowl, beat together the lemon zest, sugar and eggs until the mixture turns a very light shade of yellow and doubles in volume. If you remove the beaters from the bowl, the mixture should fall in a continuous stream.
Gently fold in the flour mixture.
Next, fold in the vanilla extract and brown butter (do not scrape the saucepan- you don't want to add all of the burnt milk solids to the madeleine batter).
Cover the mixture and chill for a minimum of 1 hour or overnight.
In the meantime, preheat the oven to 465 F or 240 C.
Then prepare the madeleine pan. Butter the madeleine tray, then place the tray into the refrigerator for 5 minutes. When the first coating of butter has solidified, brush another layer of butter atop the madeleine cups. Then, dust the tray with flour, ensuring that the cups are completely covered. While this process is a little lengthy, it is the best way to ensure ease when extracting the Madeleines from the pan. If you do not have a madeleine tray, you can use a mini muffin pan.
Scoop 2 tablespoons sized mounds of the madeleine mixture into each madeleine cup.
Place the tray into the oven for a total of 6 minutes, then reduce the temperature of the oven to 390 F (200 C) and continue cooking for 3-4 minutes or until the Madeleines are a golden brown color and form a bump on top. I like to bake the Madeleines till they are a deep golden color as I like a crisp exterior but you could bake them for a minute or so less.
Unmold the Madeleines from the tray as soon as you take them out of the oven.
Serve warm, with a cup of tea and with a dusting of icing sugar.
Madeleines taste best warm, within a day of making them- any longer and they will go from cakes to soft cookies (which are still delicious).