Step 1

For the shortbread: In a medium bowl, whisk together the flour and cornstarch.

Step 2

In a stand mixer fitted with the paddle attachment, beat together the butter, granulated sugar, and lemon zest over medium speed until creamy, 3 to 4 minutes. Scrape the bottoms and sides of the bowl and beat in the lemon juice and vanilla.

Step 3

With the mixer on low speed, gradually add the flour mixture to the butter mixture, beating until just combined after each addition.

Step 4

On a lightly floured work surface, place the dough. Divide the dough in half and form into disks. Wrap each disk in plastic wrap and refrigerate until firm, 30 minutes to 1 hour.

Step 5

Place one oven rack in the upper third of the oven and a second oven rack in the lower third of the oven. Preheat the oven to 375°F. Line 2 baking sheets with parchment paper.

Step 6

Working with 1 disk of dough at a time and keeping the second disk refrigerated, lightly flour a work surface. Roll the dough out to ¼-inch thick. Press a floured cookie stamp into the dough to make an impression. Use a 2- to 3-inch cookie cutter to cut the dough into circles. Using a thin spatula, transfer the cookies to the prepared baking sheet, spacing them about 2 inches apart. Refrigerate the scraps.

Step 7

Bake until the tops are dry and the bottoms begin to turn golden, 9 to 10 minutes. Let cool for about 5 minutes on the pan, then transfer the cookies to a cooling rack to cool completely. Repeat with the remaining dough and scraps, rerolling only once for smooth cookies.

Step 8

For the glaze: In a medium bowl, combine the powdered sugar, melted butter, and 1 tablespoon lemon juice. Add the remaining lemon juice, 1 teaspoon at a time, until the icing falls off the whisk in a fluid stream and the fallen ribbon of icing holds its shape for only a few seconds before sinking back into the bowl of icing. If the glaze is still not thin enough, add water, 1 teaspoon at a time.

Step 9

Working with one cookie at a time, use a pastry brush to brush a thin layer of glaze onto each cookie. If the pastry brush is leaving behind too many streaks or the glaze feels too thick, whisk in another teaspoon of water.

Step 10

Allow the glazed cookies to set for 15 minutes. Store at room temperature in an airtight container for up to 1 week.