Peel, core, and cut apples into chunks.
Put them in a medium saucepan or pot together with ¼ cup of water, and on the stove at low heat.
Add the lemon peel and sugar if using. Mix.
Cover and let come to a simmer. If you don't have a lid for the saucepan you're using, mix frequently over medium heat until it comes to a simmer.
Uncover and let cook until the apples are very tender. Make sure you don't run out of water so the apples don't stick or scorch to the bottom of the pan. Add a few tablespoons more water if needed. The mixture will foam a lot and that is fine.
Pierce the apples with a fork and remove them from the stove when they are very soft.
Process them with an immersion blender in the same saucepan or use a potato masher or fork for chunkier applesauce. See Notes below for other methods.
If, after you process it, it's too thin, you can return it to the stove (low heat) and cook it further, stirring constantly, until you get to the desired consistency.
Let the hot applesauce cool down for a while before transferring it to an airtight jar and refrigerating for a week or so. For canning it, see the Top Tips section in the post.
Freezing: after it's completely cooled down, you can use freezer bags or an appropriate airtight container and freeze the applesauce. It'll last several months.