Combine granulated sugar and fresh mint leaves in a food processor; pulse until the mint is pulverized and the sugar is a uniform light green.

In a separate bowl, whisk together the flour, baking soda, and salt until aerated and combined.

Using a stand mixer with a paddle attachment, cream the softened butter and mint-infused sugar on medium-high speed for 3 minutes until light and fluffy.

Incorporate the egg, lime zest, and rum extract into the butter mixture, beating until fully emulsified.

Reduce mixer speed to low and gradually add the dry flour mixture, mixing only until no dry streaks remain.

Cover the dough and refrigerate for 45 minutes to prevent excessive spreading and to deepen the herbal infusion.

Preheat oven to 350°F (175°C) and line two heavy-gauge baking sheets with silicone mats or parchment paper.

Measure 1.5-ounce portions of dough (approx. 2 tablespoons), roll into smooth spheres, and place 2 inches apart on the sheets.

Bake for 9 to 11 minutes; cookies should be set at the edges but remain pale and soft in the center.

Transfer to a wire rack to cool completely.

Whisk confectioners' sugar and lime juice in a small bowl until smooth; drizzle over cooled cookies and allow glaze to set for 15 minutes before serving.