Place the sugar and lemon zest in a food processor and pulse until the zest is finely chopped and the sugar is moistened, about 12–15 pulses.
In a large bowl, beat the lemon sugar and softened butter together on high speed for about 3 minutes until creamy. Scrape down the sides and bottom as needed.
Add lemon juice and vanilla extract to the butter mixture, mixing until well combined.
Gradually add flour, cornstarch, and salt, beating on low speed for about 1 minute, then on medium speed until the dough begins to clump together, about 1–3 minutes. The dough will be thick but soft.
Roll the dough between two pieces of parchment paper or onto a floured surface to about ¼ to ½ inch thick. Use extra flour if the dough is sticky.
Transfer the rolled-out dough (still on parchment paper or baking mat) onto a baking sheet. Cover and refrigerate for at least 3 hours, or up to 2 days.
Preheat the oven to 350°F (177°C) and line baking sheets with parchment paper or silicone baking mats.
Cut the chilled dough into shapes with a cookie cutter. Gather and re-roll dough scraps to continue cutting. If dough softens, refrigerate for 15 minutes before baking.
Optionally, sprinkle the cookies with coarse sparkling sugar before baking.
Bake for 14–15 minutes, or until the edges are set and just beginning to turn golden. Smaller cookies may bake quicker, so keep an eye on them after 10 minutes. Allow to cool on the baking sheet for 10 minutes before transferring to a wire rack.
For the icing: In a bowl, whisk together sifted confectioners' sugar, lemon juice, milk, and vanilla until smooth.
Once cookies are cool, dip the tops into the icing, letting any excess drip off before placing them on a cooling rack to dry. Garnish with fresh lemon zest if desired.
Notes
These lemon shortbread cookies are a refreshing, melt-in-your-mouth treat. Feel free to add a dash of coarse sugar on top for a sparkly finish or enjoy them plain.