Fill the boiling-water canner, set the rack in, and start heating.
Wash the apples, remove the stems, then quarter and core them. No need to peel.
Cook the apples slowly in the cider and vinegar until soft.
Press the fruit through a food mill, colander, or strainer to make a smooth pulp.
Cook the pulp with both sugars, the cinnamon, and the cloves, stirring frequently, until thick. Optional: do this cook-down in a slow cooker on LOW for about 8-10 hours, with the lid ajar and occasional stirring (OSU Extension).
Sterilize half-pint or pint jars: cover them with hot water to 1 inch above the tops and boil 10 minutes. Above 1,000 ft, add 1 minute per additional 1,000 ft. Quart jars need not be presterilized.
Test for doneness: a spoonful held away from the steam for 2 minutes stays mounded, or no rim of liquid separates around a spoonful on a plate.
Fill the hot butter into the hot jars, leaving 1/4-inch headspace. Remove air bubbles, wipe the rims, and put on the lids and bands.
Process in a boiling-water canner (hot pack). Half-pints or pints: 5 min at 0-1,000 ft, 10 min at 1,001-6,000 ft, 15 min above 6,000 ft. Quarts: 10, 15, and 20 min.
Cool the jars undisturbed on a towel for 12-24 hours, then check the seals. Refrigerate any jar that didn't seal.