Step 1

Combine sugar and lemon zest in a food processor and pulse for 1 minute, until zest is finely ground. If you don’t have a food processor, beat sugar and lemon zest in a large mixer bowl on low speed until well mixed.

Step 2

Beat butter, lemon-sugar, vanilla, and salt in a large mixer bowl at medium speed until soft and creamy, about 2 minutes. Scrape down bowl as needed. Reduce speed to low and mix in flour just until incorporated and dough begins to come together in large pieces. Finished dough will be somewhat crumbly but will hold together when pressed.

Step 3

Divide the dough in half and turn out each half onto a piece of plastic wrap. Roll each half into a 6 1/2-inch log, using the plastic wrap to help shape the soft dough. Wrap logs tightly in plastic wrap and chill for 1 hour in the refrigerator (see note below).

Step 4

Line two baking sheets with parchment paper. Place granulated sugar on a large plate and roll the logs of dough to coat generously with sugar. Using a sharp knife, cut logs into 1/2-inch rounds. Place rounds 2 inches apart on baking sheets. If desired, gently press rounds with the bottom of a measuring cup to make them perfectly level. Chill for 20 minutes in the refrigerator while preheating oven to 350°F.

Step 5

for 14 to 15 minutes, rotating pans halfway through, until light golden and edges are set. Cool for 5 minutes on baking sheets, then transfer to a wire rack to cool completely.

Step 6

Whisk together melted butter, salt, lemon zest, powdered sugar, lemon juice, and milk until smooth. Dips the tops of cookies in lemon glaze (or drizzle with glaze) and let set before serving.

Step 7

Store cookies at room temperature in an airtight container.