Melt the butter in the microwave or on a stovetop.
In a medium sized bowl: add eggs and sugar. Use a hand held mixer, beat on high speed for 5 minutes (until consistent).
Add vanilla extract, lemon zest and beat on medium speed for 1 minute.
Add ½ cup of sifted flour, salt, baking powder and mix. Continue with the rest of the flour and mix.
Slowly pour melted butter in the bowl. Using a whisk and mix until everything is combined (do not recommend using a mixer for this step). Refrigerate for 30 minutes.
Preheat the oven to 375°F (190°C).
Smear softened butter on madeleine pan. Scoop out about 1 ½ tbsp batter in the center of the shell (no need to spread to the edges). You should have 18 cookies total.
Bake for 10-13 minutes or until lightly golden brown. Remove the cookies from the oven and let them rest on the baking sheet for 2-3 minutes, then transfer to the cookies to a wire rack and let cool completely.
Optional: Sprinkle with powdered sugar.