Rinse 2 pounds of fresh damsons under cool water and remove any stems. Place the whole damsons in a large, heavy-bottomed pot or Dutch oven.
Add 1/4 cup of water to the pot with the damsons. Bring the mixture to a simmer over medium heat, then reduce heat to low, cover, and cook for about 20 minutes until the fruit is very soft and the skins have split.
Remove the pot from the heat. Carefully press the cooked fruit through a fine-mesh sieve or food mill set over a large bowl to remove the pits and skins, extracting as much pulp as possible. You should have about 4 cups of puree.
Return the strained damson puree to the clean pot. Stir in 4 cups of granulated sugar and 1 tablespoon of fresh lemon juice until the sugar begins to dissolve.
Cook the mixture over medium heat, stirring constantly, until the sugar is completely dissolved. Add 1/2 teaspoon of unsalted butter to help reduce foaming.
Bring the jam to a full rolling boil that cannot be stirred down. Cook for about 15-20 minutes, stirring frequently to prevent sticking, until the jam reaches 220°F on a candy thermometer or passes the plate test (a spoonful placed on a chilled plate wrinkles when pushed).
Remove the pot from the heat and skim off any foam from the surface with a spoon. Let the jam rest for 5 minutes to allow the fruit to distribute evenly.
Ladle the hot jam into sterilized half-pint jars, leaving 1/4 inch of headspace. Wipe the rims, apply lids and bands, and process in a boiling water bath for 10 minutes if canning, or allow to cool completely before refrigerating.