Beat icing sugar and butter until creamy.
Beat in the lime and zest.
Sift the flour and cornstarch and salt together. Beat in to the butter mixture until uniformly mixed.
Chill the dough one hour.
Preheat the oven to 350 degrees. Spread out your oven racks to top and bottom thirds. Prepare a cookie baking sheet with parchment paper or silpat.
Remove dough from the fridge. Take about a ½ tablespoon of dough and form it into a ball. The dough is very stiff and somewhat prone to crumbling but the ball will come together from the heat of your hand as you roll it.
Place the ball on the cookie sheet, flatten slightly and press our thumb into the center of the dough to make an indentation. Repeat till all the dough has been formed into cookies.
Bake 12-15 minutes. Rotate your baking sheets halfway through. The cookies will not spread or brown.
When cookies are done remove from the oven and allow them to cool completely before icing.