Put the mint leaves and 2 of the lime wedges in a large cocktail shaker and muddle them with a muddler or a long-handled wooden spoon. Add the lime juice, simple syrup, rum, and ice and stir 40 times with a long-handled swiveling cocktail spoon or the wooden spoon. Add the sparkling water and salt and stir well.
For each drink, fill a low cocktail glass half full with ice, add a mint sprig and a wedge of lime, and pour over the cocktail mixture. Stir and serve ice cold.
For this recipe, Ina used a simple syrup made from a 2 to 1 ratio of sugar to water.