1. ​​Line a large rimmed baking sheet (or two smaller baking sheets) with parchment paper and set aside. 
2. To a stand mixer fitted with the paddle attachment (or use a mixing bowl and a hand mixer), add the sugar and lemon zest. Mix together until the lemon zest is incorporated into the sugar.
3. Add the butter, lemon juice, and vanilla; mix again until creamed together. 
4. Add the flour, baking soda, and salt. Mix until the dough comes together, about 1 minute.
5. Scoop the dough from the mixing bowl and form into a ball with your hands. Place the dough ball on a lightly floured surface and, using a rolling pin, roll it out to about ¼-inch thickness. 
6. Use a 2 ½ inch cookie cutter to cut the dough into circles and place on the prepared baking sheet. Gather up the dough scraps and roll them into a ball before rolling the dough out to ¼-inch thickness again. Repeat until you have 16 cookies and all of the dough has been used. 
7. Place the baking sheet(s) with cookies in the refrigerator for 30 minutes to chill.
8. Preheat the oven to 350°F while the cookies chill. Once the chill time is up, remove the baking sheet from the refrigerator. Place the baking sheet on the middle rack in the preheated oven. If using more than one pan, bake only one at a time, leaving the second pan in the fridge while the first one bakes.
9. Bake until very lightly golden on the bottoms, 7 to 9 minutes. Remove from the oven and let the cookies cool on the baking sheet for 5 minutes before transferring to a cooling rack. Let cool completely before adding the icing.

To make the icing:

1. To a small bowl, add the powdered sugar, 2 teaspoons lemon juice and 1 teaspoon milk; whisk until smooth. If too dry, add additional milk, ½ teaspoon at a time, until desired consistency is reached.
2. Spread the icing on top of each cookie to within 1/8th inch to the edge. Sprinkle with additional fresh lemon zest, if desired.
3. Store iced cookies in an airtight container at room temperature for up to 4 days or in the refrigerator for up to 7 days. To freeze, place the unfrosted cookies in a freezer-safe container or bag and freeze for up to 3 months.