Combine apples and apple cider in a heavy pot over medium heat. Cover and cook 20-25 minutes, stirring occasionally, until completely soft.
Use an immersion blender directly in the pot, or pass through a food mill, to achieve a smooth puree.
Stir in the sourdough discard - it blends seamlessly into the warm puree and begins adding its tang immediately.
Add sugar, cinnamon, cloves, allspice, lemon juice, and salt. Taste and adjust sugar to your liking.
Reduce heat to the lowest setting. Cook uncovered, stirring every 10-15 minutes, for 1 1/2 to 2 1/2 hours, until thick and dark and holding its shape on a cold spoon.
Ladle into sterilized jars while hot. Refrigerate up to 3 weeks, or water-bath can (10 minutes) for 12 months shelf-stable.