Mix water, apple cider, apple juice concentrate, white sugar, brown sugar, and cinnamon sticks together in a large saucepan; bring to a boil. Reduce heat and simmer until mixture is reduced, about 45 minutes. Remove from heat and cool for 15 minutes. Stir grain alcohol into mixture until evenly combined. Cool 'apple pie liquor' completely. Pour apple pie liquor and 1 cinnamon stick into a jar or container.