Preheat oven to 350°F (175°C). Line baking sheets with parchment paper.
Cream butter, granulated sugar, and powdered sugar until light and fluffy.
Beat in egg, lime juice, lime zest, vanilla, and rum extract if using.
In a separate bowl, whisk flour, baking powder, baking soda, and salt.
Mix dry ingredients into wet ingredients until just combined.
Fold in finely minced mint.
Scoop 1½-tablespoon portions onto baking sheets, spacing 2 inches apart.
Bake 10–12 minutes, until edges are set and bottoms are lightly golden.
Cool on the baking sheet for 5 minutes, then transfer to a wire rack.
Finish with extra lime zest or drizzle with lime glaze once cooled.