For the cookies: Preheat oven to 350 degrees. In a medium mixing bowl stir together flour, baking soda, salt, and lemon zest, set aside.
In a large mixing bowl with an electric mixer on medium speed cream butter and sugar until fluffy. Beat in egg, vanilla, and lemon juice. On low speed gradually add the flour mixture and mix just until blended.
Drop dough by rounded tablespoonfuls 2 inches apart onto ungreased baking sheets. Bake for 15 to 20 minutes. Transfer cookies to a cooling rack to cool completely.
For the glaze: In a small mixing bowl wisk together powdered sugar, lemon juice, and lemon zest. Dip tops of each completley cooled cookie into glaze. Place cookies on cooling rack until glaze is set.