Make the cookie dough: Add the sugar and lemon zest to a food processor or blender and pulse at least 12 times.
In a large bowl, use a handheld mixer or a stand mixer to beat the lemon zest sugar and butter on high for 3 minutes. Scrape the sides if needed, then add the sourdough discard, lemon juice, and vanilla, and mix again until combined. Add the flour, cornstarch, and salt and mix on medium until the dough begins to clump together.
Lightly flour a sheet of parchment paper and turn the dough out onto it. Use a floured rolling pin to roll the dough out on the parchment until it is about 1/2" thick evenly. Transfer the dough sheet with the parchment onto a baking sheet, cover it with plastic wrap, and chill the dough for at least 3 hours, or up to 5 days in the fridge.
After chilling is complete, preheat the oven to 350°F and line baking sheets with parchment paper or silicone baking mats.
Use a cookie cutter to cut the dough into your preferred shape. Shoot for a 2-3" diameter cookie. You can also use a knife to cut even-sized squares. Re-roll any excess dough until all the dough is used up.
Bake cookies in the preheated oven for 14-15 minutes, until the edges are set and the cookies are barely turning golden. Cool cookies on the baking sheet for 10 minutes and then transfer them to a cooling rack to cool completely.
Optional Icing: Whisk all the icing ingredients together in a small bowl. Let the cookies cool completely and then dip the tops into the icing and let the excess icing drip off before setting the cookie on a wire rack. If you'd like, sprinkle each cookie with fresh lemon zest while the icing is still wet.Let the cookies dry on the rack until the icing has set completely. You should be able to press your finger on the icing without leaving any indentation. The icing will set in about an hour, and then you can transfer or stack the cookies in an airtight container.