Wash the plums. Slice them in half, and remove and discard the pits and any stems.
Chop the plums into 1-inch pieces and place them in a large, non-reactive bowl (e.g. glass or plastic). Stir in the sugar.
Cover and refrigerate the plum mixture for 12-24 hours, stirring the mixture once or twice during this time.
Transfer the plum mixture (including all the accumulated juices) to a large, heavy-bottomed saucepan.
Cut a 4- to 5-inch square of cheesecloth.
Using the back of a spoon or side of a knife, crush the cardamom pods to expose the seeds inside. Then, place the crushed pods/seeds in the center of the cheesecloth.
Using a small paring knife, slice the vanilla bean in half lengthwise. Scrape out the seeds and add them to the saucepan with the plum mixture.
Chop the spent vanilla bean pod into smaller pieces and add it to the cheesecloth with the cardamom.
Gather up the corners of the cheesecloth to create a pouch. Then, securely tie it off with a piece of kitchen string or twine. Add the pouch to the plum mixture.
Place the saucepan over medium-high heat and bring the mixture to a simmer, stirring occasionally.
As the mixture thickens, reduce the heat and stir more frequently to prevent the bottom from scorching. As you stir, use your spoon to mash some of the larger plum pieces up against the side of the saucepan to help break them down.
Cook jam until it's thick and glossy, about 75-90 minutes. Remove the saucepan from the heat and set it aside to cool.
The jam will thicken slightly more as it cools. Remove and discard the cheesecloth pouch.
Once cooled, transfer the jam to clean re-sealable containers and store in your refrigerator for 2-4 weeks, or in your freezer for several months.