In a 2-quart saucepan, combine 3 cups of the apple juice, the cinnamon stick, and the bay leaf and bring to a boil. Add the celery root and cook for 15 minutes, until the liquid is reduced to a slightly thick syrup.
Using a slotted spoon, remove the bay leaf and cinnamon stick. Add the apples, the remaining apple juice, and the vinegar and cook 5 to 7 minutes more, or until the liquid is completely reduced. Spoon the mixture into the bowl of a food processor and process until smooth. Season with salt and pepper and serve.