Add the pumpkin purée, water, brown sugar, maple syrup, cinnamon sticks, pumpkin pie spice, and salt to a medium saucepan.
Bring to a gentle simmer over medium heat, stirring until the sugar has completely dissolved.
Reduce the heat to low and simmer for approx. 8-10 minutes, stirring occasionally.
Remove from the heat and discard the cinnamon sticks.
Strain through a fine-mesh strainer into a heatproof container.
Stir the vanilla bean paste or extract into the strained syrup.
Let cool completely, then transfer to an airtight container and refrigerate for up to 2-4 weeks. Shake before using.