Place 4 cups milk and cinnamon stick in a large saucepan over low heat.

While milk heats up, place remaining two cups milk and cookies in a blender container.  Blend until smooth. Transfer mixture to saucepan with milk and cinnamon.  Add piloncillo and vanilla, increase heat to medium-low and stir continuously until piloncillo dissolves; about 8-10 minutes. Continue stirring until mixture thickens; about 5 minutes.  Add more milk if thinner consistency is desired. Remove and discard cinnamon stick.  Serve hot.