For the cake batter: Preheat the oven to 350 degrees F. Line a standard muffin pan or cupcake pan with 12 cupcake liners.
Combine the buttermilk and the mint extract in a bowl and set aside. In a medium bowl, sift together the flour, baking powder and salt and set aside.
Using a stand mixer on medium-high speed, beat the granulated sugar and butter together until pale, light and fluffy. Add the eggs one at a time, mixing thoroughly after each addition; scrape down the sides of the bowl. Add the rum, vanilla and lime zest and juice and mix until combined. Add the dry ingredients and buttermilk mixture in alternating batches and mix until just combined.
Divide the batter among the cupcake liners, filling them three-quarters full. Bake until a tester inserted into the centers comes out clean, about 20 minutes. Cool the cupcakes on a cooling rack for 5 to 10 minutes.
For the mojito syrup: In a small saucepan, combine the granulated sugar, butter and 1/4 cup water over medium-high heat. Bring the mixture to a boil and cook, stirring often, until the butter is melted and the sugar is dissolved. Off heat, carefully add the rum and mix until incorporated. Add the mint extract and lime zest and mix until combined.
For the frosting: Using a stand mixer, beat the confectioners' sugar and butter together until well combined. Add the rum, lime juice and salt and mix on medium-high speed until incorporated and fluffy.
To assemble: Brush the mojito syrup onto the cupcakes with a pastry brush, or drizzle over the cupcakes using a spoon. Allow the cupcakes to cool completely and absorb the syrup before topping with the frosting.
This recipe was created by a contestant during a cooking competition. It has not been tested for home use.