Supreme 4 grapefruits (cut away the peels and pith, then cut the segments free from the membranes) and freeze the segments until solid, 8 hours or up to overnight.
Juice the remaining 6 grapefruits using an electric citrus juicer. Keep pushing until the grapefruit pulp is out, creating a cup. Reserve the "cups" to serve in. (Freeze any extras for another use.)
Combine the grapefruit juice, rum, lime juice and agave syrup in a large cocktail shaker filled with ice (see Cook's Note). Fit the reserved emptied grapefruit halves into large coupe glasses. Strain the cocktail into the grapefruit halves. Sprinkle in the grapefruit ice and garnish with mint leaves.
Start with half of the ingredients in the cocktail shaker, just in case your shaker isn't large enough to make all 6 servings at once.