In a small saucepan over medium heat, melt your butter. Continue to cook until the butter is golden brown and has a nutty fragrance. This takes about 4-5 minutes. Remember the the butter turns very quickly from golden to burnt. Remove from the heat, pour into a small bowl, and cool to room temperature.
Into the bowl of an electric mixer, place the eggs and sugar and whip on high for about five minutes until the mixture is thick and pale.
Decrease to low speed and add the milk and vanilla extract. Mix till combined.
In a separate bowl, add flour, baking powder and salt and whisk to combine. Add to the bowl of the electric stand mixer and mix on low speed until combined.
Remove the bowl from the stand mixer and add the browned butter. Using a spatular, fold the butter into the mixture. Take care not to over mix. Cover with cling wrap and refrigerate for 2-3 hours to let the batter rest (or overnight if needed).
30 minutes before baking, pre-heat the oven to 180°C/350°F. Using a pastry brush, generously brush melted butter into the moulds of two Madeleine pans, and dust with a little flour (knocking out the excess). Place pans in the fridge until ready to use.
Once the oven is at temperature, remove the batter from the fridge and one Madeleine pan. Fill the deepest part of each well with 1 level tablespoon of batter. Do not level the batter out. Remove the other pan from the fridge and fill it in the same way.
Place in the oven and bake for 10-12 minutes, check after 8 minutes and rotate the pans. They are ready when the ‘humps’ have risen, and they‘re browned on the edges. Remove from the oven and allow to cool on a wire rack for a couple of minutes before removing them from the pans and allowing them to cool. (Tip: give the pans a little jiggle, and they should pop out. If they stick, run a knife around the edges to loosen them.)