Using a muddler or large wooden spoon, mash sugar and herbs together in a large saucepan until the herbs are bruised and fragrant.Advertisement
Add water and bitters; bring to a simmer over medium-high heat, stirring often, about 5 minutes. Stir in rum and cook for 1 minute. Add butter and cook, stirring, until it melts.
Strain the mixture into a pitcher. Let stand for about 10 minutes to mellow the flavors. Serve in cordial glasses or other small heat-safe glasses, garnishing each drink with a mint leaf.
Make Ahead Tip: Let cool, cover and refrigerate for up to 1 day. Reheat on the stovetop over low heat or in the microwave on Medium.
1 other carbohydrate, 1 alcohol equivalent