Combine apple quarters, cinnamon sticks and water in a large stockpot. Cook over low heat, stirring occasionally, until apples are very tender, about 25 minutes.
Process apples in batches through a Harvest Pro, food mill or food processor until smooth, removing cinnamon sticks. Alternately, crush apples with a potato masher for a chunkier texture.
Prepare boiling water canner. Heat jars in simmering water until ready to use, do not boil. Wash lids in warm soapy water and set aside with bands.
Return applesauce to stock pot, add maple syrup and bring to a simmer over medium heat, stirring constantly, for 10 minutes.
Ladle hot applesauce into a hot jar leaving a ½ inch headspace. Remove air bubbles. Wipe jar rim. Center lid on jar and apply band, adjust to fingertip tight. Place jar in boiling water canner. Repeat until all jars are filled.
Process jars 25 minutes, adjusting for altitude. Turn off heat, remove lid, let jars stand 5 minutes. Remove jars and cool 12-24 hours. Check lids for seal, they should not flex when center is pressed.