In a large bowl, combine sugar and lemon zest, using your fingers to rub them together until mixture is fragrant, about 30 seconds. Transfer ¼ cup lemon-sugar mixture to a small bowl. Set aside.

Add butter to the large bowl with lemon-sugar mixture and, using an electric mixer, beat on medium-high speed until fluffy and combined, about 2 minutes, stopping to scrape down sides as needed.

Add egg, lemon juice, and vanilla; beat on medium-high speed until creamy and fluffy, about 15 seconds.

In a medium bowl, whisk flour, baking powder, and salt to combine. Add flour mixture to butter mixture, and beat on low speed until just combined, about 20 seconds.

Divide dough in half, about 14 ounces (397g) each. Place each half on a large piece of plastic wrap. Fold plastic wrap over to cover dough, and, using your hands, roll each portion of dough into a smooth 2-inch-wide log about 6 inches long and wrap tightly, twisting the ends of the plastic to seal. Refrigerate until firm, at least 1 hour and 30 minutes or up to 8 hours.

Adjust oven rack to middle position and preheat oven to 350°F (175ºC). Line 2 large rimmed baking sheets with parchment paper. Remove logs from refrigerator. Spread the reserved lemon-sugar on a large plate. Remove dough logs from plastic wrap, and roll in lemon-sugar mixture to coat evenly, pressing firmly to help sugar adhere evenly.

Using a sharp knife, trim off about ¼ inch from each end of dough log and discard or reserve for another use. Using a sharp knife, slice each log into ½-inch thick rounds; you should have about 12 cookies per log.

Place slices on prepared baking sheets, leaving at least 1 ½ inches between each cookie. Freeze until firm, about 15 minutes.

Bake cookies, 1 sheet at a time (leaving second sheet in the freezer while first bakes), until edges are set and beginning to lightly brown, 14 to 15 minutes. Cool on baking sheets for 5 minutes. Transfer to a wire rack, and let cool to room temperature, about 30 minutes.

Prepare the Glaze: In a medium bowl, whisk powdered sugar, lemon juice, and vanilla extract until smooth and well combined.

To Assemble: Using an offset spatula, spread the lemon glaze (about 1 teaspoon per cookie) on the cooled cookies and garnish with lemon zest. Let stand until the glaze is set, about 15 minutes.