Heat the pumpkin puree in a saucepan with the pumpkin spice over medium-high heat. In a minute, add the maple syrup and stir. Reduce the heat to medium low. Once the maple syrup starts to bubble, mix in 1/2 cup of milk and stir until the milk is warmed up. There's no need for the milk to bubble. Turn off the heat.
Pour the coffee into a mug and then add the pumpkin milk.
Pour the rest of the milk into a jar. Make sure that there's plenty of space left in the jar to help aerate the milk as you shake it. Put the lid on the jar, and shake it for about a minute. You should be able to see a lot of frothy bubbles forming. Microwave the jar of milk uncovered for 30 seconds. You should now see some nice frothy milk on the top. Spoon the milk froth on top of the latte. Repeat this step by shaking and microwaving the same milk if you want more froth in the drink. I repeated this step once.