Preheat oven to 350 degrees Fahrenheit. Line cookie trays with a silicone baking mat or parchment paper.
In a mixer with the paddle attachment, cream the butter with the sugars until smooth. Then add the oil, lime juice, lime zest, and eggs. Mix well.
In a separate bowl, combine the flour, baking soda, cream of tartar, and salt. Add all at once to the wet ingredients and mix until everything comes together. It should be slightly crumbly and not sticky at all.
Using a #30 cookie scoop (about 2 Tbsp of cookie dough) scoop dough balls onto a silicone or parchment lined baking sheet (no more than 8 per tray).
Put 1/4 cup granulated sugar on a small plate. Dip the bottom of a glass cup into the sugar, and press it into the center of each dough ball until about 1/2-inch thick. TIP: Rocking the glass in a circular motion flattens the cookies more evenly.
Bake at 350˚F for 7-8 minutes. Do NOT over-bake! The tops will look matte instead of glossy when they are finished. Cool on tray for several minutes before transferring to a wire rack to cool completely.
With an electric hand mixer and a large bowl, beat the butter and coconut oil together until smooth.
Mix in the powdered sugar one cup at a time, adding the coconut extract and coconut creamer after the first two cups. Then add the remaining powdered sugar. Mix thoroughly. Add extra coconut creamer (or milk) 1/2 Tbsp at a time, if needed, until desired consistency.
Spread frosting over cooled cookies and top with a lime wedge. Squeeze the lime wedge juice over the frosting before consuming. Enjoy! Store extra cookies in an airtight container at room temperature or in the refrigerator.
Notes
HOW TO CUT LIME WEDGES: Cut the lime in half width-wise down the middle of the fruit (like you would do if you were juicing it). Then cut each half into thirds (at about a 45 degree angle) from the center of the fruit, for six lime wedges.TO FREEZE: Lay cookies flat onto a cookie tray, putting wax paper between each layer. Wrap tightly with plastic wrap. Freeze frosting separately for best results. Re-whip the thawed frosting, adding extra milk or coconut creamer if needed.