1In a bowl of an electric mixer, fitted with the whisk attachment, mix butter and 1/3 cup of confectioners' sugar on medium speed, until pale and fluffy. Add lemon zest, lemon juice and vanilla. Mix until fluffy.
2Whisk together, flour, cornstarch, and salt in a bowl. Add to butter mixer on low speed until just combined.
3Divide the dough in half. Place each half on a sheet of parchment paper. Roll in parchment to form a log 1-1/4 inches in diameter. Refrigerate logs until cold and firm, at least 1 hour.
4Preheat oven to 350 F. Remove parchment from logs; cut into 1/4 inch thick rounds. Space rounds 1 inch apart on baking sheet lined with parchment paper.
5Bake cookies until barely golden, about 13 minutes, rotating sheets halfway through.
Transfer cookies to wire racks and cool slighly.
While cookies are still warm toss with the remaining 2/3 cup confectioner's sugar.
Store in air tight container at room temperature for 2 weeks.