Melt the butter and vanilla together in the bottom of a heavy pot, over medium heat. Add the cinnamon stick and sugar and stir to slightly dissolve the sugar. Add the blackberries and bring the mixture to a boil, stirring frequently to make sure nothing sticks to the bottom of the pot.
Once the jam has boiled, bring it back down to a simmer and allow it to cook, uncovered, for about 20 minutes to half an hour. Stir often, breaking the blackberries up with the spoon as you go.
To test if the jam is ready, drop a spoonful onto a bowl or small plate, and place in the freezer for about ten minutes. the jam is ready if the spoonful jiggles when removed from the freezer. If it's not ready, cook another 10 minutes and test again.
When the jam is cooked, remove the cinnamon stick and transfer the contents of the pot to a glass jar. If you're canning the jam, now is the time to process the jar. If not, chill the jam, covered, for at least one hour before use.