Step 1: Wash, peel, core, and chop the apples into roughly 1-inch pieces. Smaller pieces will cook faster.
Step 2: Place the chopped apples into a large saucepan or pot. Add the water or apple cider, cinnamon, nutmeg (if using), and salt (if using). If adding sugar, you can add it now or after cooking to taste.
Step 3: Bring the mixture to a simmer over medium heat. Once simmering, reduce the heat to low, cover the pot, and cook for 15-20 minutes, or until the apples are very soft and easily pierced with a fork. Stir occasionally to prevent sticking.
Step 4: Once the apples are tender, remove the pot from the heat. For a chunky applesauce, mash the apples with a potato masher to your desired consistency. For a smooth applesauce, use an immersion blender directly in the pot, or carefully transfer the mixture to a regular blender or food processor and blend until smooth. Be cautious when blending hot liquids.
Step 5: Taste the applesauce. If desired, stir in additional sugar or maple syrup and more cinnamon until it reaches your preferred sweetness and spice level. If it's too thick, you can add a tablespoon or two of hot water to thin it out.
Step 6: Serve warm or cold. Store leftover applesauce in an airtight container in the refrigerator for up to 7-10 days, or freeze for up to 3 months.