Place plums in a large stockpot and add enough water to cover the bottom of stockpot; cook over very low heat, stirring occasionally, until plums are broken down, about 2 hours. Remove stockpot from heat and cool for 2 hours.
Return stockpot to stovetop over low heat; cook gently for 2 to 3 hours. Remove from heat and keep at room temperature, 8 hours to overnight.
Cook plums over low heat, stirring occasionally, about 2 hours. Remove from heat and let stand at room temperature for 2 hours.
Bring plums to a boil; add sugar, lower heat, and simmer until thickened, about 15 minutes.
Sterilize jars and lids in boiling water for at least 5 minutes. Pack plum butter into hot jars, filling to within 1/4 inch of the top. Run a clean knife or thin spatula around the insides of the jars to remove any air bubbles. Wipe the rims of the jars with a moist paper towel to remove any food residue. Top with lids and screw on rings.
Place a rack in the bottom of a large stockpot and fill halfway with water. Bring to a boil and lower jars into the boiling water using a holder. Leave a 2-inch space between the jars. Pour in more boiling water if necessary to bring the water level to at least 1 inch above the tops of the jars. Bring the water to a rolling boil, cover the pot, and process for 15 minutes.
Remove the jars from the stockpot and place onto a cloth-covered or wood surface, several inches apart, until cool. Press the top of each lid with a finger, ensuring that the seal is tight (lid does not move up or down at all). Store in a cool, dark area.