In a food processor work bowl, add sugar and lemon zest. Process until sugar and zest are well mixed, about 20 seconds. Scrape into a large mixing bowl. Reserve about 1/2 cup of lemon sugar in a separate bowl for later use.
Add flour, baking powder and salt into food processor and pulse briefly to mix. Add cold butter into food processor with flour mixture and process with short pulses until mixture resembles cornmeal and no large lumps of butter remain, about 10 pulses. Pour this mixture into the sugar and lemon mixture in the large mixing bowl and set aside.
In a medium bowl, mix egg yolks, vanilla extract and lemon juice together. Using an electric mixer with a paddle attachment, pour in the lemon/sugar and flour-butter mixture and stir until combined, about 15 seconds. Then, while the mixer is still running, slowly add egg/lemon juice mixture and mix until well combined and dough pulls away from the sides of the bowl, about 15 to 30 seconds.
Scrape the dough into a large bowl and refrigerate until cold, but not solid, about 30 minutes. Use an ice cream scoop to scoop the dough onto parchment-lined cookie sheets. Take the dough balls and roll the tops into the reserved lemon sugar. Use the bottom of a water glass to press the cookies flat, about 1/4 inch thick.
Bake the cookies until slightly browned, about 15 minutes. Remove the parchment paper with the cookies from the tray to a cooling rack and cool completely, about 1 hour. Once the cookies have completely cooled, ice with Lemon Icing.
Lemon Icing : In a medium bowl, whisk sugar and lemon juice together until smooth. Spoon onto cooled cookies and spread evenly with the back of a spoon. Let stand on a cooling rack until the glaze is set, about 1 hour. Yield: 3 dozen cookies