Mix the dry ingredients: In a medium bowl, whisk together the flour, cornstarch, and salt. Set aside.

Prepare the lemon-sugar mixture: In the bowl of a stand mixer fitted with the paddle attachment, combine the powdered sugar and lemon zest. Mix on low speed for about 1 minute. The lemon zest will release oils, slightly clumping the sugar.

Add butter and vanilla: Add room-temperature butter to the sugar-zest mixture. Beat on medium speed until light and fluffy, about 3 to 5 minutes. Scrape down the bowl. Add vanilla extract and mix until just incorporated.

Add the dry ingredients: On low speed, gradually add the flour mixture. Mix until just combined. Finish by stirring with a spatula to ensure no streaks of flour remain.

Shape the dough: Place the dough on plastic wrap and roll into a log about 1 ¼ inches in diameter. Wrap tightly and refrigerate until firm—at least 1 hour and up to 12 hours.

Preheat and prep: Preheat oven to 350°F (175°C). Line two baking sheets with parchment paper.

Slice and bake: Unwrap the dough and slice into ¼-inch-thick rounds. Arrange on prepared sheets about 1 inch apart. Bake for 11 to 13 minutes, rotating halfway through, until just set and slightly golden on the bottom. Cool completely on a wire rack.

Make the glaze: In a small bowl, whisk together powdered sugar, lemon zest, and 2 tablespoons lemon juice. Add more juice a few drops at a time until glaze is smooth and spreadable.

Glaze the cookies: Spoon a small amount of glaze onto each cooled cookie and spread with the back of a spoon. Allow the glaze to set for about 1 hour before serving.