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 to 30 minutes. Once finished, strain the milk into a bowl using a fine mesh sieve. The milk may appear curdled or clumpy, but don't worry, it will come back together. Press on any milk solids and mint leaves to extract all the liquid possible. Give the milk a quick stir to smooth it out and set aside.
While the mint is steeping in the milk, preheat the oven to 325 degrees F. Line a standard muffin tin with paper liners; set aside.
In a medium bowl, whisk together the flour, baking powder and salt; set aside.
In an electric mixer on medium-high speed, beat the butter and sugar together until pale, light, and fluffy (about 5 minutes).
Reduce the mixer speed to medium and add the eggs one at at time, mixing thoroughly after each addition.
Scrape the sides of the bowl and add the lime zest, lime juice, vanilla extract and rum. Mix until combined. (The mixture may start to look curdled at this point, but don’t worry, it will all come back together, power on!)
Reduce the mixer speed low. Add the dry ingredients in three batches, alternating with the buttermilk in two batches. Mix only until just incorporated, using a rubber spatula to give it one last mix by hand.
Divide the batter between the muffin cups. Bake for approximately 25 minutes or until just slightly golden and a skewer shows only moist crumbs attached, rotating the pan at the halfway point.
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 using the whisk attachment for 5 minutes. Reduce the speed to medium-low and gradually add the powdered sugar, mixing and scraping the sides of the bowl until all is incorporated. Give it a mix on medium speed for about 30 seconds. Add the lime juice and rum and mix on medium-high speed until incorporated and fluffy. If the frosting appears a bit too soft, add some additional sugar, one spoonful at a time until desired consistency is reached. Frost cupcakes and garnish, if desired, with a lime wedge and sprig of fresh mint.