Preheat the oven to 325°F and line one or two large baking sheets with parchment paper. Set aside.
In a large mixing bowl or the bowl of a stand mixer, combine the powdered sugar, basil and lemon zest. Beat together for two minutes or until the sugar is fragrant.
Whisk in the flour until combined.
Grate the frozen butter and transfer to the mixing bowl. Beat in on low speed until evenly distributed and combined with the dry mixture.
Use the dough hook to incorporate the lemon juice and then gradually add water, 1 to 2 teaspoons at a time, until a soft dough forms. (If you don't have a dough hook, beat in on low speed.)
Transfer dough to one of the prepared baking sheets and place another piece of parchment paper on top. Use a rolling pin to roll the dough out to about ¼ inch thick.
Use 2 to 3 inch cookie cookers to cut out cookies. Place the cookies on the other baking sheet, making sure to leave cookies about one inch apart.
Sprinkle each cookie lightly with coarse sugar and bake for 12-17 minutes or just until the edges are golden brown.
Cool the cookies for 5 minutes on the baking sheet before transferring to a cooling rack. Let cookies cool completely before storing or enjoying.
Garnish with additional lemon zest and fresh basil, if desired.