Preheat a grill to high. Brush the papaya slices with olive oil and put on the grill. Grill for 3 minutes per side and then transfer to a cutting board. Cut the papaya into large chunks. Fill 2 bucket glasses each with 4 pieces of lime, grilled papaya chunks, 1 tablespoon of raw sugar, and muddle the ingredients. Wrap whole ice cubes in a kitchen towel and pound the ice with a rolling pin, cocktail muddler, or heavy pan to create crushed ice. Fill the glasses with crushed ice and add 1-ounce of cachaca to each glass. Using a bar spoon, give all ingredients a few spins and add additional crushed ice. Garnish each drink with 1 slice of grilled papaya, a mint sprig, and several papaya seeds.