To a bowl add flour, corn starch, baking powder and salt. Stir and set aside.
Zest the lemon(s). Depending on size, you may only need to zest one. You need two teaspoons for the dough. If your lemon didn't render enough, zest the second.
Juice the lemon(s). You will need 3 tablespoons for the glaze. Set aside.
In a separate bowl cream together the softened butter and sugar with a hand mixer or in a stand mixer. Mix on high for 3 minutes until light and fluffy.
Add the lemon zest, egg and vanilla. Continue mixing until fully incorporated. Scrape the bowl down, mix again.
Slowly add the dry ingredients into the wet on low. Allow to fully incorporate for a minute until the dough comes together.
Split the dough in half as it's much easier to work with in a smaller batch.
Dust your bottom piece of parchment paper with flour and add half the dough. Dust the top of the cookie dough with flour and cover with a second piece of parchent paper. Roll out the dough to 5/8" thick between the parchment paper.
If you don't have a crescent cookie cutter use a glass to cut out circles in the dough. Once your circles are cut out and excess dough is removed, use the glass to create the crescent shape.
Re-roll out the excess dough between the parchment paper, dusting with flour as needed. Continue creating your crescents until you're out of dough.
Place the cutouts in the refrigerator to chill for 20 minutes while you work on the second half of dough. Do not chill the cookies on the sheet pan they'll bake on or it will take longer to bake.
Preheat oven to 350 degres.
Place chilled crescents on a parchment lined sheet pan an inch apart. They shouldn't spread out too much.
Bake for 14-15 minutes. If they got too cold during chill time they may need a little more time. Keep your eye on it. You don't want the bottoms browning. The shine should be gone from the tops when they're done.
Bake one sheet pan at a time on the middle rack.
Allow the cookies to cool to room temperature.
While the cookies cool make your lemon glaze by whisking powdered sugar and freshly squeezed lemon juice to a bowl. Whisk until all lumps are gone and it's a smooth consistency.
Dip cookies in the glaze, allow excess glaze to run off. Place on either wire racks or parchment paper to harden.
Dust with powdered sugar.