In a stand mixer fitted with a paddle attachment, cream together the softened unsalted butter, granulated sugar, and brown sugar until the mixture is light, fluffy, and well-aerated.
Beat in the egg, fresh lime zest, minced mint, rum extract, and spearmint extract until fully combined.
In a separate bowl, whisk together the flour, baking soda, and salt.
Gradually add the dry ingredients to the wet ingredients, mixing on low speed until just combined. Do not over-mix, as this will affect the pillowy texture.
Cover the dough and chill in the refrigerator for at least 30 to 60 minutes. This step is crucial to prevent the cookies from spreading too thin.
Preheat your oven to 350°F (175°C) and line baking sheets with parchment paper or a Silpat mat.
Using a 2-ounce cookie scoop, portion the dough into balls and place them on the prepared baking sheets.
Bake for 10-12 minutes, or until the edges are just beginning to turn golden but the centers remain soft and chewy.
Allow the cookies to cool on the baking sheet for 5 minutes before transferring them to a wire rack to cool completely.
If desired, whisk together powdered sugar and lime juice to create a glaze and drizzle over the cooled cookies. Garnish with fresh lime zest and mint just before serving.