1. In a small saucepan, bring the lychee syrup and ginger to a boil. Remove from the heat, cover and let steep for 30 minutes.
2. Meanwhile, stuff 16 lychees with brandied cherries. Set each lychee in the cup of a mini muffin pan of in an ice cube tray. Strain the lychee-ginger syrup and pour it over the lychees. Freeze until firm, at least 4 hours.
3. Meanwhile, in a pitcher, stir the sugar into the lime juice until dissolved. Add the rum and refrigerate until chilled, about 1 hour.
4. Unmold the lychee ice cubes into a small punch bowl. Pour in the lime-rum mixture. Add the ginger beer, stir gently and serve in ice-filled glasses.
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