In a medium size sauce pan, whisk the sugar, lime zest, and lime juice.
In a small bowl, whisk the whole eggs, and egg yolk.
On medium to high heat, and while whisking, slowly add the eggs to the lime/sugar mixture.
Whisk constantly and let the mixture cook for about 4-5minutes, depending on how hot your stove gets. Once it starts to bubble, I let it bubble for about 2 minutes, stirring constantly.
Remove from the heat and whisk in the butter. It might take a few minutes. I throw my butter in the mixture, let it sit for a couple of minutes, and then whisk it after it melts. (If you are adding food coloring, this is the time to add it. Remember, a little goes a long way)
Strain through a fine-mesh sieve into a glass or plastic Tupperware bowl. With a rubber spatula, push the lime mixture through the sieve.
Press plastic wrap onto the mixture to help prevent a skin from forming and let it sit out until it comes to room temperature, and then refrigerate at least 4 hours to set.
For the cookies
Preheat the oven to 325 degrees F. Line 3 sheet pans with parchment paper. Aside.
In the bowl of a stand mixer fitted with the paddle attachment cream the butter, oil, granulated sugar and powdered sugar on medium speed.
Add the eggs, one time at a time mixing after each addition. Cream on medium speed for 30 seconds after each egg is added. Remember to scrape the bottom and the sides of the bowl with a rubber spatula. Add in the vanilla, sour cream and coconut extract. Mix just until combined.
Add the baking soda, baking powder, salt, and the flour. Mix on the lowest speed until no flour streaks remain and the mixture is evenly combined; don’t overmix.
Scoop the dough into about 3-tablespoon sized portions and roll into balls. Place several inches apart on the prepared baking sheets. (I baked 9 cookies on each sheet pan.) Using a flat surface, like a pancake spatula, or the bottom of a cup that has a flat surface, slowly press down each. I like mine a little thicker, so I didn’t smash down very much. (Ideally the cookies should be around ½ thick)
Bake the cookies for 10-12 minutes until just set.
Let the cookies cool for a few minutes on the baking sheets before removing to a cooling rack to cool completely.
While the cookies are cooling make the coconut frosting
For cream cheese frosting
In the bowl of an electric mixer, combine the butter and cream cheese. Cream on medium speed for about two minutes. Scrape down the sides and bottom of the bowl. Beat again for 2 more minutes.
With the mixer on low speed, add the powdered sugar, followed by the, and salt. Beat on low just until mixed. Then turn the mixer to medium high speed and beat the mixture for 2-3 minutes.
Add the vanilla, lime curd, and coconut extract and mix on low just until combined.
Once the cookies are completely cooled, and no longer warm, frost the with a small offset spatula and evenly spread to the edge of the cookie.
Use the tip of the offset spatula, slowly swirl some of the lime curd on top of the cream cheese frosting. Top with lime for decor and enjoy! (Note: my favorite way to eat these cookies is when the cookie completely chilled in the fridge. I highly recommend trying it)