In the bowl of a stand mixer or a large bowl, cream the ⅔ cup granulated sugar, the powdered sugar and butter until fluffy and light. Add the vegetable oil, lime juice and zest and the egg and cream until smooth.
In a smaller bowl, toss together the flour with with baking soda, cream of tartar and salt. Mix half of the flour mixture to the wet ingredients and then add the other half and mix until dough comes together and is mixed through.
Roll dough into balls the size of golf balls (I use a #24 cookie scoop) and place 2 inches apart on a baking sheet lined with parchment paper or a Silpat or sprayed with non-stick baking spray.
Add the remaining 3 tablespoons sugar to a shallow bowl. Smudge a bit of the cookie dough on the bottom of a small glass jar before dipping in the sugar so the dough doesn’t stick to the jar. Press the dough balls with the jar dipped in sugar to ½ cm thick and the edges crack just a bit.
Bake for 8 minutes. The cookies won’t brown and may look look undercooked, but they will be done.
Let the cookies sit on the baking sheet for a few minutes before transferring to a cooling rack.
While the cookies are baking, prepare the frosting by combining the ingredients in a bowl and mixing until smooth.