Heat milk over medium-high heat in a medium saucepan to 180° or until tiny bubbles form around edge (do not boil). Remove from heat; keep warm.
Combine sugar and 1 tablespoon water in a large, heavy saucepan over medium-high heat, and cook until sugar dissolves, stirring as needed to dissolve sugar evenly (about 2 minutes). Continue cooking for 2 minutes or until golden (do not stir).
Carefully add hot milk to caramelized sugar in a slow, steady stream (mixture will bubble vigorously), stirring until sugar dissolves. Add bananas. Cover, reduce heat, and simmer 5 minutes. Remove from heat; cool 10 minutes.
Place half of milk mixture, ice cream, rum, salt, and 1/8 teaspoon nutmeg in a blender. Remove center piece of blender lid (to allow steam to escape); secure blender lid on blender. Place a clean towel over opening in blender lid (to avoid splatters). Blend until smooth. Pour into a large pitcher. Repeat procedure with remaining milk mixture. Cover and refrigerate 6 hours or up to overnight. Top with whipped topping and grated nutmeg, if desired.