Begin by melting the unsalted butter in a saucepan over medium heat. Allow it to cook until it turns a golden brown, giving it a nutty aroma. Remove from heat and let it cool slightly.
In a mixing bowl, whisk together the eggs and granulated sugar until the mixture is pale and thickened. This can take about 5 minutes. Add in the vanilla extract and lemon zest, and whisk until combined.
In another bowl, sift together the all-purpose flour, baking powder, and salt. This will ensure there are no lumps and that the baking powder is evenly distributed.
Gently fold the dry ingredients into the egg mixture until just combined. Be careful not to overmix. Then, add the browned butter, folding it in gently until the batter is smooth.
Cover the batter with plastic wrap and refrigerate for at least 30 minutes. Chilling the batter helps develop the flavors and improves the texture of the madeleines.
Preheat your oven to 375°F (190°C). Grease your madeleine pan with butter or non-stick spray.
Spoon the batter into each madeleine mold, filling each cavity about 3/4 full. This allows room for the madeleines to rise and take on their signature shape.
Bake in the preheated oven for 10-12 minutes, or until the edges are golden brown and the centers spring back when lightly pressed.
Once baked, remove the madeleines from the pan and transfer them to a cooling rack. If desired, dust with sifted confectioners sugar before serving.