If desired, peel the apples before removing the cores. Cut the apples into large wedges or chunks.

In a large, heavy-bottomed pot, combine the apple pieces with the cinnamon stick, lemon juice, vanilla extract, and star anise. Pour in the water and gently stir to coat the apples.

Cover the pot with a lid and set it over medium-low heat. Cook, stirring occasionally, until the apples become very soft and begin to break down, about 25 to 35 minutes. Check the pot periodically to ensure there is enough moisture—if the apples begin sticking, add a splash of water and slightly lower the heat.

Once the apples are fully softened, remove the pot from heat and allow the mixture to cool for a few minutes. Taste the applesauce and, if desired, add a sweetener, starting with a teaspoon and adjusting to taste.

For a smooth applesauce, blend the mixture using an immersion blender or pass it through a mesh strainer. If you prefer a chunkier texture, mash the apples with a fork or potato masher. If the apple skins were left on, a food mill fitted with a medium disk will easily separate them from the sauce.

The applesauce will continue to thicken as it cools. If it appears too watery, return it to the stove and let it simmer gently until it reaches the desired consistency.