In a small saucepan over medium-low heat, melt half a cup of butter and set aside, allowing it to cool slightly.

In a large mixing bowl, beat the eggs, sugar, lemon zest, and vanilla on medium speed for 2-3 minutes, until the mixture becomes fluffy and pale.

In a separate bowl, whisk together 1 cup of flour, baking powder, and salt. Gently fold half of this dry mixture into the egg mixture until it’s mostly combined, then add the remaining flour mixture. Take half of the prepared batter and fold it into the melted butter, ensuring it’s just incorporated, then fold this buttery mixture back into the remaining batter until a glossy, smooth consistency is achieved. Cover the bowl and refrigerate for 30 minutes.

Preheat the oven to 350°F (175°C). In a small microwave-safe bowl, melt the remaining 2 tablespoons of butter for 30-60 seconds, then blend in the remaining tablespoon of flour. Use a pastry brush to thoroughly coat the wells of a madeleine pan with this butter-flour mixture.

Fill each well of the prepared pan with 1 tablespoon of the chilled batter. Bake for 10-12 minutes, or until the tops of the madeleines spring back when gently pressed. Allow them to cool briefly before inverting the pan to release the madeleines. Place them on a wire rack to cool completely, then dust with confectioners' sugar before serving.