Combine the sugar and cinnamon and divide the mixture equally among each of the grapefruit halves. Broil for 4 to 5 minutes, or until the sugar and juice boil and the sugar starts to darken. Serve each grapefruit half in a small bowl, and pour any excess liquid on top of each portion. If desired, splash with a little extra rum before serving.
Cut each grapefruit in half across its equator. With a grapefruit knife or paring knife, cut the fruit away from the rind around the circumference of the fruit. Section each segment by running the knife from the center along the membrane to the rind. This will loosen the fruit, making it easier to eat. Place the grapefruit halves in a casserole dish or baking pan.
Drizzle 1 tablespoon of rum over each grapefruit half. Poke a few holes in the flesh if the rum does not soak in.