Melt the 1/2 cup butter gently over the stove in a small pan, and set aside to let cool. You can also melt in in the microwave, but be warned, butter right out of the fridge likes to explode in the microwave, making a huge mess and wasting butter!
In the bowl of a stand mixer with the whisk attachment, beat the eggs and sugar together on high for ~8-10 minutes. Make sure to do the full time, even if it feels long! This is what gives the madeleines their beautiful texture. Beat in the lemon zest and vanilla until combined. Remove the bowl from the mixer, the rest will be mixed by hand.
In a separate bowl, whisk together the flower, baking powder, and salt. Gently fold the flour mix into the egg and sugar with a spatula. I recommend mixing the flour in a bit at a time, and be gentle with it, it may seem like it's not fully incorporating but it will get there!
Take ~1/4 cup of the batter and add it to the melted butter and mix until it is fully combined, and then add all that back to the batter and mix gently until incorporated. The batter will be thick and shiny, and very delicate so be careful!
Cover the bowl and let the batter chill for at least 30 minutes, but no more than 60 or else it will get too firm. Preheat the oven to 350°F.
Melt a few tbsp of butter and using a pastry brush (or a paper towel in a pinch, which is what I used) grease the madeleine pan lightly. Spoon a large tablespoon of batter into each scalloped well. You don't need to spread it out at the edges, and make sure not to over fill, they'll spread and rise quite a bit in the oven.
Bake for 10-12 minutes. They're done when the tops are slightly springy when pressed. Remove the pan and carefully invert it, and the madeleines should fall right out. Let them cool on a wire rack.
At this point, you can do whatever you like with them! Dust with sugar, dip in chocolate (see below), etc. They start to get stale after a day or so, but if you can't eat them all in time (which will be easy to do) they're great to dip in coffee in the morning!
Enjoy!
