Melt the butter in the microwave or stovetop and set it aside so it can cool to room temperature.
Using a mixer fitted with the whisk attachment beat the eggs and sugar together on high speed for 8 minutes. The mixture will look thicker and pale light yellow. It should clink to the whisk slightly when done.
Whisk in the lemon zest and vanilla extract until combined.
Remove bowl from the mixer.
Whisk the flour, baking powder, and salt together in a small bowl.
Using a large spoon or spatula gently fold half of the flour into the egg mixture. Be very gentle with the batter when mixing.
Gently mix the other half of the flour with the egg mixture.
Pour melted butter into a different mixing bowl. Stir 1/4 cup of the batter into the melted butter. Stir slowly and gently until incorporated together.
Add the rest of the batter to the butter mixture and stir gently until combined.
Cover the batter and chill in the refrigerator for 30 minutes, any longer and the dough will not flow into the madeleine pan easily and the butter will set up.
Preheat the oven to 350 while your dough chills in the fridge
Melt the remaining 1 tablespoons butter. Using a pastry brush, lightly brush the pan with melted butter. It's important to avoid the cookies from sticking.
The batter will be quite airy and spongey after resting. Carefully spoon 1 tablespoon of dough into each scalloped center of the pan. Try not to deflate the dough. You will not need to spread it all the way to the edges, the heat from the oven will warm the dough and it will slide into the creases.
Bake for 10-12 minutes. The madeleines are done when the tops spring back after lightly pressed with your finger. Remove from oven and carefully flip the pan over onto the counter. Transfer the warm madeleines to a wire rack to slightly cool.
Dust with confectioners’ sugar, if desired, before serving.