Core and chop the apples, leaving the skin on. In a large lidded pot, heat the apples with the honey, vanilla, and cinnamon sticks over low heat. Continue to cook with the lid on until the apples are soft, stirring every 5 to 10 minutes, about 30 minutes. If the sauce is too runny at the end, cook for several minutes with the lid off to cook off some of the juices.
Allow to cool, then remove the cinnamon and puree the applesauce in a food processor or a blender. Applesauce can be refrigerated in an airtight container for a week or frozen for up to 3 months.