Beat butter and sugar until lightened in color and texture. Add egg, vanilla, lemon zest, and lemon juice, and stir until combined, then beat again another 30 seconds. Add salt, baking powder, and cornstarch and mix until well incorporated. Add flour and stir until just combined.
Use a standard size cookie scoop to scoop cookies onto a sheet lined with a silpat or parchment paper. Place a little sugar on a plate and use the bottom of a glass to dip into the sugar then flatten each cookie to a little more than 1/2 inch thick.
Bake at 350 degrees for 9-11 minutes, just until the tops start to crack a little.
Let cool on baking sheet for 2-3 minutes then place on a cooling rack.
Make the glaze while cookies cool. Add powdered sugar to a small bowl. Add 2 tablespoons lemon juice and use a fork to stir until combined.
1 cup powdered sugar, 2-3 tablespoons lemon juice
Once cookies cool for at least ten minutes, scoop out a bit of the icing for each cookie and spread gently across the top. Let the icing harden for two to three hours, then store in a tightly sealed container on your counter. Separate layers with parchment paper to ensure the glaze doesn't stick to the cookie above it.