In a small saucepan over medium heat, whisk together the milk, pumpkin puree, and maple syrup.
Heat the mixture while whisking constantly until it reaches approximately 160°F (71°C), ensuring it does not reach a boil.
Remove the saucepan from heat and stir in the pumpkin pie spice and vanilla extract.
Optional: Process the mixture with a handheld milk frother or high-speed blender for 20 seconds to create a foam texture.
Pour the hot coffee or espresso into a large mug.
Pour the prepared pumpkin milk over the coffee and serve immediately.