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Peel, core, and chop apples; set apple chunks aside, reserving peels and cores. Combine peels, cores, and 2 cups apple cider in a saucepan and bring to a boil. Reduce heat to low, cover, and simmer 30 minutes. Meanwhile, combine chopped apples, sugar, remaining cider, and cinnamon stick in a large, non-aluminum saucepan and bring to a boil. Reduce heat to low. Cover and simmer, stirring occasionally, until apples begin to fall apart, 20 to 40 minutes. Press peel-core mixture through a food mill or sieve placed over a bowl to catch juice. Reserve juice and discard peels and cores. Add juice to cooked apples in the saucepan and return mixture to a boil. Reduce heat and simmer, uncovered and stirring occasionally, until thick, 40 to 60 minutes. Remove and discard cinnamon stick. Stir in pumpkin. Preheat the oven to 300 degrees F (150 degrees C). Pour apple-pumpkin mixture into a large, shallow roasting pan. Place roasting pan into the preheated oven, leaving door slightly ajar. Bake, stirring every 15 minutes, until thick enough to spread, 1 to 1 1/2 hours. Inspect 6 half-pint jars for cracks and rings for rust, discarding any defective ones. Immerse in simmering water until apple-pumpkin butter is ready. Wash new, unused lids and rings in warm soapy water. Spoon apple-pumpkin butter into sterilized jars leaving 1/2-inch head space. Wipe rims and threads clean; top with hot lids and screw bands on firmly. Place a rack in the bottom of a large stockpot and fill halfway with water. Bring to a boil and lower jars 2 inches apart into the boiling water using a holder. Pour in more boiling water to cover jars by at least 1 inch. Bring to a rolling boil, cover, and process for 5 minutes. Remove the jars from the stockpot and let rest, several inches apart, for 12 to 24 hours. Press the center of each lid with a finger to ensure the lid does not move up or down. Remove the rings for storage and store in a cool, dark area.