Preheat your oven to 350°F. Line a baking sheet with parchment paper or a silicone baking mat and set aside.

In a small bowl, whisk together the flour, baking soda, baking powder, and salt. Set aside.

In the bowl of a stand mixer fitted with the paddle attachment—or using a large mixing bowl and a hand mixer—combine the softened butter, oil, 2⁄3 cup granulated sugar, and powdered sugar. Beat on medium speed for about 2 minutes, or until the mixture is light and fluffy.

Mix in the sour cream, egg, lime zest, lime juice (about 3 tablespoons from 1 large lime), vanilla extract, and coconut extract. Stir just until combined.

Gradually add the dry ingredients to the wet mixture, mixing on low speed until the dough is just combined. Be careful not to overmix.

Scoop about 3 tablespoons of dough for each cookie and roll into a ball. Place on the prepared baking sheet, spacing each cookie about 2 inches apart.

Dip the bottom of a drinking glass in the remaining granulated sugar, then gently press each dough ball to about ½ inch thick.

Bake for 10 minutes, or until the edges are set but not browned. The centers may look slightly underbaked—this is perfect for a soft cookie.

Remove the tray from the oven and allow cookies to cool on the pan for 5 minutes before transferring to a wire rack to cool completely.

In a medium mixing bowl, combine the softened cream cheese and butter. Beat with a hand mixer or stand mixer for 2 minutes, or until smooth and well blended.

Scrape down the sides of the bowl. Add the powdered sugar, lime juice, lime zest, and coconut extract.

Whip the frosting for 1 to 2 minutes, or until light, creamy, and easy to spread.

Once the cookies are completely cool, spread a generous layer of frosting over the tops. Serve immediately or store as directed.