1For smooth applesauce, core and coarsely chop the apples into 1/2-inch pieces, leaving on the peel. For chunky applesauce, peel, core, and coarsely chop the apples into 1/2-inch pieces. Put the apple chunks in a large, heavy-bottomed saucepan.
2Add the apple cider, juice, or water and the cinnamon stick (if using). Bring to a gentle boil over medium heat.
3Lower the heat to medium low and gently boil for 20 to 30 minutes, stirring occasionally with a wooden spoon and checking to make sure the apples are not scorching.
4When the apples are extremely tender and can be broken down to a chunky sauce with vigorous stirring, remove from the heat. Remove the cinnamon stick (if used). Let cool for a few minutes, then add the sugar, a large pinch of salt, and the lemon juice.
5For smooth applesauce, let the apples cool to room temperature. Using a food mill fitted with a medium disk, pass the apple mixture through the mill to form a smooth purée. For chunky sauce, beat the apples with a wooden spoon until the desired consistency is reached. Let cool.