In the bowl of a stand mixer fitted with a whisk attachment (a hand mixer works too), combine eggs, vanilla and salt on high speed until pale in color, about 2 minutes. With the mixer running, gradually add sugar and continue to whip on high speed for 8 more minutes. Batter is done when it holds a ribbon when the whisk is lifted.
Use a rubber spatula to fold the flour into the batter in thirds (1/4 cup at a time). Fold gently just until incorporated. Add lemon zest and pour melted butter around the edges of the bowl so it doesn't deflate the batter. Again, fold in gently making sugar to scrape the bottom of the bowl.
Cover the bowl and chill the batter for 45 minutes and up to 4 hours.
Remove the batter from the refrigerator and divide the batter evenly among a madeleine pan with 16 molds. Bake for 12 minutes. Remove madeleines from the pan, cool slightly and dust with confectioners' sugar.