In a separate bowl, combine the sugar and lemon zest until crumbly and aromatic. 

Using an electric mixer on medium speed, cream butter until light and fluffy. Then mix in the sugar mixture. Add the eggs, one at a time, followed by the lemon juice and vanilla. Mix well.

With the mixer on low speed, gradually add the flour mixture to the butter mixture, mixing just until a soft dough forms.

Using a tablespoon of dough at a time, roll the dough into balls and place on the prepared baking sheets, space them about 2 inches apart.

Bake one pan at a time for 15 minutes, or until the edges are lightly golden.

Cool the cookies on the baking sheets for a few minutes before transferring them to a wire rack to cool completely.

Whisk together the confectioners' sugar and 2 tablespoons of lemon juice until the mixture is smooth and a good consistency for drizzling. Add small amounts of lemon juice until the glaze is the desired consistency.

Once the cookies are completely cooled, drizzle the glaze over the tops of the cookies using a spoon or a piping bag. Sprinkle a more lemon zest and/or poppy seeds over the glaze if desired. Allow the glaze to set before storing or serving.