Bring mint leaves, water, and 1 cup of the sugar to a boil in a small saucepan over medium-high; boil 1 minute. Remove from heat, and let stand 30 minutes. Chill until cold, about 1 hour. (Mint mixture can be stored in refrigerator up to 7 days.)
Preheat grill to medium (350°F to 400°F). Sprinkle each side of watermelon slices with 1/2 teaspoon of the sugar. Place on grill grate lightly coated with cooking spray; grill, uncovered, until grill marks appear, 2 to 3 minutes per side. Remove from grill. Cool slightly, and cut into 1-inch cubes. Arrange cubes on a single layer on a baking sheet, and freeze until solid, at least 2 hours or overnight. Transfer cubes to a zip-top plastic freezer bag, and store in freezer until ready to use.
Process 2 cups frozen watermelon cubes, 2 cups ice cubes, rum, lime juice, and 1/4 cup mint simple syrup in a blender until smooth. Stir in club soda. Pour into 4 stemmed cocktail glasses, and garnish with a mint sprig.
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