Place an everyday spoon into the freezer for later.
Place the diced pumpkin flesh, granulated sugar, and water in a pot.
Simmer over medium heat for one hour, stirring occasionally.
At the end of the first hour, add the lemon zest, vanilla bean, cinnamon sticks, and rum, to the mixture. Stir until incorporated.
Cook over medium heat for an additional hour, stirring occasionally, and scraping the sides of the pot so that the jam does not stick.
When you are nearing the end of the full 2 hours, test for doneness. Spoon out a small amount of jam, and drop it onto a separate, frozen spoon and allow it cool. Wipe your finger through the cooled jam. It is ready when it does not run back together after being wiped.
When the jam is done, remove the cinnamon sticks and vanilla bean, and place the jam into jars.
Allow the jars of jam to cool to room temperature before putting them away. They can be stored in your pantry for up to 2 years. Refrigerate after opening.