Melt the butter in a heavy-bottomed saucepan and cook over low heat for about 10 minutes until browned. Pour into a small bowl and allow to cool while you prepare the rest of the batter.
In a medium bowl whisk together the flour, baking powder, and salt. Set aside.
In a large bowl add the eggs and sugar. Using either a stand mixer or hand mixer, use the whisk attachment to beat the eggs and sugar on medium-high speed until very thick and pale, about 5-6 minutes. Add the vanilla and lemon zest and fold in using a spatula.
Fold in the dry ingredients, mixing until just combined. Fold in the butter.
Cover the bowl with plastic wrap, ensuring it does not touch the batter, and refrigerate overnight.
About 20 minutes before you’re ready to bake, preheat the oven to 375°. Prep your madeleine pan by brushing the pan with melted butter and lightly dusting it with flour.
Scoop a generous tablespoon in the center of each madeleine well. If the batter is still quite thick, gently press it a little flatter into the well.
Bake for 9-10 minutes or so, until the bellies have risen and are turning golden brown. The edges should just be starting to brown.
Let the madeleines cool in the pan for a few minutes before gently loosening them from the pan. Let them finish cooling on a cooling rack.