Preheat oven to 325°F (163°C) and line a baking sheet with parchment paper.
In a mixing bowl, rub the lemon zest into the granulated sugar using your fingers until fragrant.
Add the softened butter and beat with a mixer for 1–2 minutes until smooth. Do not overmix or whip air into the mixture.
Mix in the lemon juice and vanilla extract until fully combined.
Add the flour and salt (and cornstarch if using). Mix until a soft dough forms. The dough should be slightly crumbly but hold together when pressed.
Shape the dough into a log about 2 inches thick. Wrap and chill in the refrigerator for 30–45 minutes.
Slice the dough into 1/4-inch thick rounds and place on the prepared baking sheet, spacing slightly apart. Sprinkle with coarse sugar if desired.
Bake for 15–18 minutes, until the edges are lightly golden and the tops look dry and set.
For extra crispness, turn off the oven and leave the cookies inside for an additional 5 minutes.
Allow cookies to cool on the baking sheet for 5 minutes, then transfer to a wire rack to cool completely. Cookies will crisp up fully as they cool.