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Preheat oven to 350 degrees. In a medium sized mixing bowl, combine flour, cornstarch, baking soda, cream of tarter, and salt.
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Set aside.
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In a large mixing bowl, cream together butter, coconut oil, sugar (1¼ cups), and powdered sugar.
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Stir in the 2 tablespoons lime juice, 2 teaspoons lime zest, and 1 teaspoon coconut extract. Mix in eggs.
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Slowly add in flour mixture until combined. Roll the dough into golf sized balls and place onto a non stick cookie sheet or use a parchment line sheet.
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To give the cookies their signature rough edge, dip the bottom of a glass into the reserved ¼ cup sugar and press
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onto the cookie ball to flatten the cookie. You still want the cookie to be thicker so don't press them too thin.
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Bake for 8-10 minutes or until the bottom is lightly brown. The tops will not brown. The cookies stay soft so do not over bake. Cool completely.
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To make the coconut lime cream cheese frosting: Beat cream cheese, butter, lime juice, lime zest, 2 1/4 teaspoons coconut extract, and vanilla together until smooth and fluffy. Add confectioners' sugar. Beat until creamy. Frost the center of each cookie. Garnish with additional lime zest if desired.
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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.