To make mint simple syrup, in a small saucepan bring ¼ cup of water to a boil. Add ¼ cup of sugar and stir until dissolved. Cook for additional 3 minutes or so, or until the syrup is slightly thickened.
Throw in half of mint leaves and let it sit for at least 15 minutes.
In a medium bowl, add remaining 1 tablespoon of sugar and remaining 10 mint leaves.
Bruise the mint leaves with a muddler.
Stir in mint simple syrup, lime juice and both rums.
Pour the mixture into a pitcher with ice and add club soda right before serving.