Cut the butter into tablespoons and melt it in a heavy saucepan over medium heat; once melted, reduce heat to low and cook until the milk solids sink and turn golden brown, about 10 minutes.
Remove the browned butter from heat and pour into a small bowl; stir in the vanilla and lemon zest and set aside to cool.
In a medium 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 on medium while gradually adding the granulated sugar; then increase to medium-high and whip until very thick and pale, about 3–4 minutes.
Turn off the mixer and use a silicone spatula to fold the flour mixture into the whipped eggs in three additions, mixing only until combined.
Fold the cooled browned butter mixture into the batter until just incorporated.
Cover the bowl and refrigerate the batter for at least 4 hours, or up to 2 days.
About 30 minutes before baking, preheat the oven to 375°F (190°C). Generously brush madeleine molds with butter and lightly dust with flour.
Using a medium cookie scoop, place level scoops of batter into the center of each mold; do not spread the batter.
Bake for about 12 minutes, until the madeleines have risen and are golden brown.
Cool the madeleines in the pan on a cooling rack for 1–2 minutes, then gently loosen them from the molds, using a small knife if they stick.
Dust with sifted confectioners' sugar, if using, and serve warm.