Peel bananas and cut into about 1/2-inch slices.
Place slices slightly apart in an 8-to 9-inch cake pan.
Cover and freeze until banana is frozen, at least 2 hours or up to 1 week.
Cover and chill coffee until cold, at least 30 minutes.
In a blender or food processor, combine banana, coffee, and milk.
Whirl until pureed; add sugar and rum to taste.
Pour into chilled glasses and sprinkle with cinnamon.