In a nonstick pan over medium heat add the butter. Once the butter melts add the brown sugar and let the sugar melt and become bubbly. Add the bananas and cook until caramelized, about 2 minutes per side. Let cool slightly.
Mix the ice, ice cream, milk, vanilla and bananas in a blender until smooth but still thick. Pour the shake into 4 chilled glasses and sprinkle with the cinnamon.