Cut the butter in tablespoons and place them in a heavy saucepan over medium heat. Once the butter has melted, reduce the heat to low and continue cooking until the solids sink to the bottom of the pan and turn golden brown, about 10 minutes. Remove from heat and pour the browned butter into a small bowl.
Stir the vanilla and lemon zest into the browned butter, then set aside to cool.
In a medium mixing bowl, whisk together the flour, baking powder, and salt. Set aside.
In the bowl of a stand mixer fitted with the whisk attachment, beat the eggs at medium speed while gradually adding the sugar. Increase the speed to medium-high and continue whipping until very thick and pale, about 3 to 4 minutes.
Fold in the flour mixture in three additions, stirring just until combined, then fold in the butter mixture.
Cover the bowl and refrigerate for at least 4 hours, up to 2 days.
30 minutes before baking, preheat the oven to 375 degrees (F). Generously brush the madeleine molds with butter and lightly dust with flour.
Spoon level scoops of the batter into each mold. Bake for 12 minutes, or until the “bellies” rise and are golden brown.
Cool in the pan on a cooling rack for a few minutes, then gently loosen from the molds. Dust with confectioners’ sugar and serve warm.