In a large bowl with an electric mixer, beat the eggs and the sugar until thick and pale, about 6 minutes.
In a separate bowl, whisk together the flour and baking powder. Sift 1/3 of the flour mixture over the egg mixture, using a spatula to carefully fold and combine. Then add the remaining mixture in 2 additions in the same fashion.
Take a large spoonful of the batter and add this to a small bowl along with the butter, mixing them together to lighten the butter. Pour the butter mixture into the batter and gently fold together to combine. Press a sheet of plastic wrap directly onto the surface of the batter and then refrigerate for at least 3 hours before baking. (The mixture can, blissfully, be stashed in the fridge for up to 2 days at this stage.)
Make the lemon glaze
In a medium bowl, use a spoon to mix together the lemon juice, zest, and confectiners' sugar until you have a smooth, pourable glaze. Add more lemon juice or sugar as needed. Press a piece of plastic wrap directly onto the surface of the glaze and let stand at room temperature until needed.
Bake the madeleines
About an hour before baking, butter a full-size or petite madeleine pan very well and dust with a little flour, tapping out any excess. Transfer the pan to the freezer to chill.
Preheat the oven to 425°F (220°C). When you're ready to bake, spoon the batter into the madeleine molds, being careful not to fill any of the molds more than 3/4 full. You don’t need to spread the batter, as this will happen in due course as the madeleines bake. Bake until the cakes are cooked through and the edges have started to brown, 8 to 10 minutes for full-size madeleines and 4 to 5 minutes for petite madeleines.
Remove the pan from the oven and immediately turn the madeleines out onto a wire rack. Let cool for 10 minutes.
Place the wire rack over a sheet of parchment paper. When the madeleines are cool, dip them into the glaze, coating them fully, letting the excess glaze to drip back into the bowl before placing the madeleines on the rack to set, about 20 minutes. These are best served as close as possible to coming out of the oven and are most spectacular the day they're baked.
Notes
Madeleine variations
Madeleines are really versatile. The author has flavored them in all sorts of ways, from mixing cacao nibs into the batter to using a blood orange glaze and even dipping them in tempered chocolate.