Preheat your oven to 350°F (175°C). Line baking sheets with parchment paper or silicone baking mats.
In a medium bowl, whisk together the flour, baking powder, baking soda, and salt. Set aside.
In a large bowl, using an electric mixer, cream together the butter, granulated sugar, and brown sugar until light and fluffy, about 2-3 minutes.
Beat in the egg and egg yolk, one at a time, ensuring each is fully incorporated before adding the next. Mix in the lime zest, lime juice, chopped mint, vanilla extract, and rum extract (if using).
Gradually add the dry ingredients to the wet mixture, mixing until just combined. Be careful not to over-mix the dough.
Scoop out tablespoon-sized portions of dough and roll them into balls. Roll each ball in granulated sugar to coat.
Place the dough balls on the prepared baking sheets, spacing them about 2 inches apart. Bake for 10-12 minutes until the edges are lightly golden but the centres are still soft.
Allow the cookies to cool on the baking sheets for a few minutes before transferring them to a wire rack to cool completely.