Preheat the oven to 325 degrees. Fill muffin tin with paper liners and set aside. Combine the buttermilk and the ½ cup mint leaves in a small saucepan over medium heat. Warm until just starting to steam (do not simmer), then remove from the heat and cover. Let steep for 15 minutes. Once finished, strain the milk into a bowl using a fine mesh sieve. When we made these, the buttermilk separated when warmed and had chunks of mint in it causing us a little angst, but once incorporated into the mix, we were good with it! In a medium bowl, whisk together the flour, baking powder and salt; set aside. With an electric mixer on medium-high speed, beat the butter and sugar together until pale, light, and fluffy. Reduce the mixer speed to medium and add the eggs one at at time, mixing thoroughly. Scrape the sides of the bowl and add the lime zest, lime juice, vanilla extract and rum. Mix until combined. Reduce mixer to low speed. Add other ingredients, alternating the buttermilk mixture and the dry ingredients.
Divide the batter between the muffin cups. Bake for approximately 25 minutes or until just slightly golden.
While the cupcakes are baking, combine the rum and mint sprigs in a small saucepan and warm over medium heat until the rum is heated through (do not let it simmer). Remove the pan from the heat, cover, and let steep until ready to use.
Allow cupcakes to cool for 5 to 10 minutes and then remove to a cooling rack. Brush the tops of the cupcakes with the mint-infused rum. Set the cupcakes aside to cool completely before frosting them.
To make the frosting, whip the butter on medium-high speed of an electric mixer for 5 minutes. Reduce the speed to medium-low and gradually add the powdered sugar, mixing and scraping the sides of the bowl. Add the lime juice and rum and mix on medium-high speed until incorporated and fluffy. Frost cupcakes sprinkle with a little lime zest, and garnish with a lime wedge and sprig of fresh mint.
