Preheat grill.
Once the grill is hot, grill lemon and lime slices until cooked through and grill marks appear, about 2 to 3 minutes per side.
Divide lemon and lime slices evenly between 4 highball glasses. Make sure to reserve 4 lemon slices for garnish.
Divide mint and sugar evenly between the glasses and muddle with the citrus slices.
Add rum, stirring until sugar is dissolved.
Top with ice and fill with soda water.
Garnish with extra lemon and lime slices, as well as mint sprigs.