Combine the apples, sugar and 1/2 cup water in a medium saucepan. Bring to a simmer over medium-high heat, stirring occasionally. Reduce the heat to medium; cover and cook until the apples are very soft and most of the liquid has evaporated, about 20 minutes (or microwave in a large microwave-safe bowl, covered, about 10 minutes).
Add lemon juice to taste to the applesauce, then transfer to a blender or food processor and puree (or puree in the pan with an immersion blender).
Transfer the applesauce to a bowl and refrigerate until cool.
Apricot: Stir in 1/4 cup apricot jam.
Raspberry: Add 1 cup raspberries while the applesauce is still warm; mash slightly.
Cranberry-Spice: Add 1 tablespoon brown sugar, 1/4 teaspoon ground cinnamon and 1/2 cup dried cranberries while the applesauce is still warm.