In a large bowl, beat butter and powdered sugar till creamy. Stir in flour, cornstarch, and salt till well combined.
Divide dough into two equal portions. Roll out into two logs that are 1" in diameter. (Chill shortbread dough for a few minutes if it is too sticky.)
Wrap logs in sheets of plastic wrap and chill for at least two hours, or up to three days.
When ready to bake the cookies, heat oven to 350 degrees.
Cut the logs into 1/4-inch slices and place on an un-greased cookie sheet an inch or two apart. (I line mine with silpat mats.) Bake at 350° for about 8-10 minutes or till edges just start to brown. Cool completely on a wire rack.
Frosting: Beat butter, cream cheese, and salt in a small bowl till smooth and creamy. Add lime zest and enough powdered sugar to make a spreadable frosting.
Spread frosting on top of each cookie.
Notes
You can store cookies at room temperature for up to a week in an airtight container. Place parchment or wax paper between layers.