Melt the butter in a small saucepan over low heat.
Beat the eggs with the sugar in a bowl until pale, about 5 minutes. Add the lemon zest and juice and the vanilla. Stir in the flour and baking powder, and then the melted butter.
Butter and flour two 12-cookie madeleine molds. (Traditionally, these have the shape of small shells.) Pour the batter into the molds, filling them three-quarters of the way. Refrigerate for at least 2 hours.
Preheat the oven to 425 degrees F (220 degrees).
Bake the madeleines for 5 minutes, then turn down the oven temperature to 350 degrees F (180 degrees C) and bake until golden brown and puffed in the center, about 10 minutes more.
For the lemon glaze:
Mix the confectioners' sugar and lemon juice in a bowl. Before serving, dip the top of each madeleine in the lemon glaze. Garnish with lemon zest.
Letting the batter rest in the refrigerator and then baking them at a high temperature will help the madeleines rise.