Preheat the oven to 400 degrees F. Lightly butter a bundt pan.
In a dry saute pan set over medium heat, toast the walnuts until light golden brown, about 5 minutes; cool and chop.
Melt the butter in a large skillet over medium-high heat. Add the sausage and cook, breaking up any clumps with a wooden spoon, until brown, about 5 minutes. Reduce the heat to medium, add the celery, onion, apple and 1/2 teaspoon salt and cook, stirring occasionally, until softened, 8 to 10 minutes; remove from the heat.
In a large mixing bowl, add the stuffing mix, sage, thyme and walnuts. Mix together the stock, apple butter and egg in a separate bowl, then add it to the mixing bowl along with the sausage-vegetable mixture. Add 1/2 teaspoon each salt and pepper and stir gently to combine thoroughly.
Transfer the stuffing mixture to the bundt pan and bake until hot and golden brown on top, 30 to 40 minutes. If the top starts to get too brown, cover with foil. Let rest for 10 minutes in the bundt pan, then unmold onto a serving platter.
Combine the apples, apple cider, sugar, lemon juice, cinnamon, allspice, ginger, cloves and salt in a Dutch oven or large, heavy pot and bring to a simmer over medium-high heat. Reduce the heat to medium and simmer, partially covered, until the apples are very tender, about 25 minutes.
Meanwhile, preheat the oven to 300 degrees F.
Puree the apple mixture using an immersion blender or in batches in a regular blender (be careful when blending hot liquids).
Transfer the Dutch oven to the oven. Bake, uncovered, stirring every 20 minutes so that the bottom does not burn, until the mixture is very thick and a rich brown caramel color, about 3 hours. Let cool completely.