Start by preparing the apples. Peel, core, and chop them into chunks. This will help them cook evenly and make a smoother sauce.
In a medium-sized saucepan, melt the butter over low heat. Once melted, add the chopped apples along with the Demerara sugar and a splash of water. The water helps to prevent the apples from sticking to the pan before their juices are released.
Stir the mixture gently to coat the apples in butter and sugar. Then, cover the saucepan and let the apples simmer over low heat for about 20 minutes, or until they are completely soft.
Remove the saucepan from the heat. If you prefer a chunky texture, you can leave the sauce as is. For a smoother apple sauce, use a potato masher or a blender to puree the apples to your desired consistency.
Stir in the lemon juice and cinnamon. Taste and adjust the seasoning, adding more lemon juice if you prefer a tangier sauce.
Serve the apple sauce warm, or let it cool down and refrigerate until needed. It can be stored in an airtight container in the refrigerator for up to a week.