In a large mixing bowl cream together both sugars, butter, egg, coconut extract, lime zest and juice.
In a separate bowl mix together flour, cream of tartar, salt, and baking soda.
Add dry ingredients to wet ingredients; mix well.
Roll dough into 2 inch balls and place about 1 inch apart on a cookie sheet.
In a small dish filled with extra sugar, dip the bottom of a flat bottomed glass in the sugar and then press down on each dough ball to flatten. Dip glass bottom back into the sugar before you flatten each dough ball. (Flatten to about 1/4 inch thickness.)
Bake for 8-10 minutes. Remove from oven and let cookies cool on a cooling rack to room temperature.
For the Lime Cream Cheese Frosting
In a large mixing bowl cream together butter and cream cheese for 3 minutes until light and fluffy.
Gradually add sugar and mix on low to incorporate. Add lime zest and juice; cream for 3- 5 minutes until light and fluffy.
Use a small offset spatula or butter knife to spread a dollop of frosting on each of your cooled cookies. Decorate with sliced fruit.