Preheat oven to 350 degrees and line a baking sheet with parchment paper.
In a large mixing bowl, cream together butter, coconut oil, sugar (1¼ cups), and powdered sugar.
Stir in the 2 tablespoons lime juice, 2 teaspoons lime zest, and 1 teaspoon coconut extract. Mix in eggs.
In a medium sized mixing bowl, combine flour, cornstarch, baking soda, cream of tarter, and salt.
Slowly add in flour mixture until combined. Scoop into golf sized balls about 2 tablespoons each (I use this scoop) and place onto prepared baking sheets.
To give the cookies their signature rough edge, dip the bottom of a glass into the reserved ¼ cup sugar and press onto the cookie ball to flatten the cookie. You still want the cookie to be thicker so don't press them too thin. It helps to wet the bottom of the glass, wipe it off, then dip it in sugar.
Bake for 6-8 minutes or until the bottom is lightly brown. The tops will not brown. The cookies stay soft so do not over bake. Cool completely.
To make the coconut lime cream cheese frosting: Beat cream cheese, butter, lime juice, 2 teaspoons coconut extract, and vanilla together until smooth and fluffy. Add powdered sugar. Beat until creamy. Frost the center of each cookie. Garnish with additional lime zest if desired.
Swig keeps their cookies in the fridge. This isn't necessary. In fact, I like mine at room temperature. These are better the next day when the frosting has softened the cookie.