Heat the water, milk, sugar and cinnamon stick in a medium sized saucepan, over medium heat. Let it boil then remove it from the oven. Remove the cinnamon stick.
In another saucepan over medium heat, melt the butter and let it boil, but not burn. Pull the pot from the stove and slowly add the polenta, little by little, until the mixture becomes compact. Then mix with the mixture of milk, adding gradually. Mix until mixture thickens. Serve as desired.