Preheat oven to 400 degrees. In a medium bowl, beat together butter and powdered sugar until light and fluffy. Stir in lemon zest, flour, and salt until well combined. Beat in as much flour as you can. Stir in any remaining flour with a wooden spoon. Do not chill dough.
Pack dough into a cookie press and press onto ungreased cookie sheets. Bake 5 to 8 minutes or until light brown. Allow to cool on baking sheets for one minute and then remove to wire rack.
Combine beaten egg, sugar, lemon zest, lemon juice, and butter in the top of a large double boiler. Stir until thick and drizzle over cookies.