To make the dough. In a large bowl, use an electric mixer to beat the sugar and butter until creamy. Mix in the lime juice and zest.
In a medium bowl, sift together the flour, cornstarch and salt. Stir the flour mixture into the butter mixture. Refrigerate for 1 hour.
When ready to bake, heat the oven to 350 degrees. Line a baking sheet with parchment paper.
Drop teaspoon size balls of dough onto the prepared baking sheet. Use your thumb to indent the tops of the cookies and create a "nest" for the frosting to sit inches Bake for 10 to 12 minutes, or until set. These cookies should not brown or spread, and will be quite fragile.
Let the cookies cool on the baking sheet before frosting.
To make the frosting, in a medium bowl use an electric mixer to beat the cream cheese with the butter, scraping the bowl several times to ensure there are no lumps. Mix in the powdered sugar and key lime juice. Frost each cookie with a dollop of the frosting.
Cookies can be stored for 3 days in an airtight container in the refrigerator.