Pulse the sugar and lemon zest in a food processor 12 to 15 times, until the zest is fine and the sugar is damp.
Beat the lemon sugar and butter on high for about 3 minutes, until creamy. Mix in 1 tbsp lemon juice and 1/2 tsp vanilla.
Add the flour, cornstarch and salt. Beat on low, then medium, until the dough clumps together.
Roll the dough between floured parchment to 1/4 to 1/2 inch thick. Chill on a baking sheet for at least 3 hours, or up to 2 days.
Heat the oven to 350°F (177°C) and line baking sheets with parchment.
Cut out shapes and re-roll the scraps. Chill the cut cookies for 15 minutes if the dough gets soft.
Bake for 14 to 15 minutes, until the edges are set and just turning golden. Cool on the tray for 10 minutes, then move to a wire rack.
Whisk the confectioners' sugar, 1 tbsp lemon juice, milk and 1/4 tsp vanilla until smooth. Dip the cooled cookie tops and let the icing set for 1 to 2 hours.