For the Cupcakes:
½ cup buttermilk
½ cup mint leaves, bruised
1½ cups all-purpose flour
1½ teaspoons baking powder
¼ teaspoon salt
½ cup unsalted butter, at room temperature
1 cup granulated sugar
2 eggs, at room temperature
Zest and juice of 1½ limes
2 tablespoons white rum
¼ teaspoon vanilla extract

To Brush the Cupcakes:
2 tablespoons white rum
2 sprigs fresh mint

For the Frosting:
1½ cups unsalted butter, at room temperature
4¼ cups powdered sugar
1½ tablespoons lime juice
3 tablespoons white rum