Gently beat eggs; reserve ¼ cup in a bowl.
Mix sugar, and 1 cup flour into reserved eggs; rest for 10 mins.
Melt and brown butter for 10 minutes; remove from heat.
Blend 1 tbsp flour and 1.5 tbsp browned butter; set aside.
Cool melted butter in an ice-water bath.
Add cooled butter, remaining eggs, salt, lemon, and vanilla.
Preheat the oven to 375°F.
Grease madeleine moulds with reserved butter-flour paste.
Divide batter into 24 portions; fill moulds.
Bake at 375°F for about 15 minutes.
Cakes should brown lightly, and firm to the touch.
Turn cakes out to cool, and dust with confectioners' sugar.
Store in the fridge or freeze in an airtight container.