When I had three children at home, we went through quite a bit of applesauce. To get the job done quickly, I would wash and quarter the apples, add just enough water to the pot or roaster pan to keep them from sticking, cover and cook (on the stove top or in the oven) until they were tender and then run them through a food mill to strain out the peels, seeds and stems. Then I would can or freeze the sauce. This method is fast, but only good for making smooth applesauce.
Lately, I have become more fond of chunky applesauce, so I'm doing things a bit differently. I also only have one child left at home, so I'm not making as much as I used to. Here's a easy method using a slow cooker. Yes, you have to peel and core the apples, but once they are in the cooker you have lots of leeway in the timing before you need to can or freeze the sauce. I can make a batch every few days this way instead of trying to do it all at once. I have one of the large, oval Rival Crockpots and I get 6 or 7 pint jars out of one batch. If you have an electric roaster, you can make an even larger quantity at one time.
I usually get the apples in the cooker while I'm preparing dinner and by the time the dishes are done, they are ready to mash and process.
Lazy Lady (slow cooker) Applesauce
Peel, quarter and core enough apples to fill your slow cooker. I like Gravensteins, but any mixture of apples will make good sauce. Add 1/4 to 1/2 cup of applejuice or water to the bottom, just enough to create some steam. Cover and cook on high for 1 to 2 hours or low for 2 to 3 hours, depending on your schedule. You can add a few cinnamon sticks too, if you like.
When the apples are tender, mash them into sauce right in the cooker, leaving some coarse chunks. I use a potato masher, but a wire whisk stirred through the pot does the trick too. Add sweetener (sugar, brown sugar, sucanat or honey) to taste, if desired. Cover and keep hot until ready to can. Fill pints and quarts leaving 1/2-inch headspace and process in a boiling water bath for 20 minutes. Or uncover and let cool , then ladle into jars or plastic containers, leaving 1-inch headspace and freeze.