Meanwhile, sprinkle 2/3 cup of the sugar in a large, deep skillet in an even layer. Cook over moderate heat without stirring until a honey-colored caramel forms, about 5 minutes; halfway through cooking, gently swirl the pan to combine the caramel and sugar. Add the banana halves cut side down and cook just until tender, about 2 minutes. Carefully turn the bananas and cook for 30 seconds. Off the heat, add the rum and swirl gently to combine. Cook over moderate heat just until the bananas are tender but not falling apart, about 1 minute longer. Using a slotted spoon, transfer the bananas to a plate.