Place the sugar and lemon zest in a small food processor, and pulse several times (about 12–15 times) until the sugar is moistened and lemon zest is finely chopped.

In a large bowl using a handheld or a stand mixer fitted with a paddle attachment, beat the lemon sugar and butter together on high speed until creamy, about 3 minutes. Scrape down the sides and bottom of the bowl as needed.

Add the lemon juice and vanilla, and beat until combined. Scrape down the sides and bottom of the bowl as needed.

Add the flour, cornstarch, and salt and beat on low speed for 1 minute as the mixture begins to combine. Turn the mixer up to medium speed and beat until the dough clumps together, about 1–3 minutes.

Dust a piece of parchment paper or silicone baking mat generously with flour. Using a floured rolling pin, roll the dough out to about ¼ to ½ inch thick.

Transfer the rolled-out dough (still on the parchment/baking mat) to a baking sheet. Cover and refrigerate for at least 3 hours, or up to 2 days.

Once chilled, preheat oven to 350°F (177°C). Line large baking sheets with parchment paper or silicone baking mats.

Using a cookie cutter, cut the dough into shapes. Re-roll the dough scraps and continue cutting until all is used.

Bake for 14–15 minutes, until the edges are set and just starting to turn golden. Allow cookies to cool on the baking sheet for 10 minutes, then transfer to a cooling rack to cool completely.

For the icing: In a small bowl, whisk the confectioners’ sugar, lemon juice, milk/cream, and vanilla together until smooth. Dip the tops of the cooled cookies in the icing, and let any excess drip off over the bowl before placing the cookies on a cooling rack or baking sheet to dry.