Add water, brown sugar cane and cinnamon sticks to large, heavy pot over medium-high heat. Bring water to boil, stirring occasionally to dissolve sugar.
Meanwhile, in medium bowl using whisk, whisk together masarica, salt and milk until completely smooth. Pour masarica mixture into pot with sugar mixture and bring liquid to boil, whisking occasionally; reduce heat to low. Stir in vanilla extract and continue cooking, whisking occasionally, until mixture is smooth and thick, about 10 minutes more. The atole is ready when the drink coats the back of wooden spoon.
Remove pot from heat. Divide atole evenly among serving mugs. Serve warm.