In a large bowl, combine the butter and ⅓ cup powdered sugar. Use an electric mixer to beat them together until light and fluffy. Add the zest, juice and vanilla; mix until fluffy.
In a medium bowl, whisk together the flour, cornstarch and salt. Add to the butter mixture; mix on low speed until combined.
Divide the dough in half. Place each half on an 8x12-inch piece of parchment paper. Roll in the parchment to form two logs 1¼-inches in diameter. Twist the end of the parchment to seal. Refrigerate for at least 1 hour.
Preheat the oven to 350℉. Line baking sheets with parchment paper. Remove the dough from the parchment, and cut into ¼-inch rounds. Space the rounds 1-inch apart on the prepared baking sheets. Bake until the cookies are barely golden, about 13 minutes.
While still warm, gently cover the cookies with the remaining ⅔ cup of powdered sugar (sugar in a bowl, and dip cookies in sugar). Place them on a cooling rack to cool completely. Store in an airtight container at room temperature for up to 2 weeks. They may also be frozen up to 3 months.
Notes
The cookies are delicate, so be careful when you cover them with powdered sugar.