Wisk together the dry ingredients: the sugar, cornstarch, cinnamon, ginger, salt, and nutmeg. Make sure there are no little lumps of cornstarch remaining.
You will be needing a total of two cans of full-fat coconut milk for this recipe. One can will be used in its entirety. For the second can:Carefully skim ½ cup from the top layer of cream. Try to not get any of the coconut water from the bottom layer in the measuring cup. (See notes.)
In a medium-sized saucepan, add the coconut milk, coconut cream, corn syrup, and pumpkin puree to the dry mixture (sugar, cornstarch, cinnamon, ginger, salt, and nutmeg). Mix together well ensuring everything is nice and smooth and mixed together. NOTE: You are not adding the vanilla until the mixture is fully cooked.
On medium/high heat, bring to a boil - stirring occasionally. Once the mixture comes to a boil reduce heat and let simmer for 10 minutes, still stirring occasionally. If the mixture ever stops simmering, turn up the heat slightly.
Remove from heat and stir in the vanilla and lemon juice
Place in the refrigerator or freezer to fully cool. This is a minimum of six hours or overnight in the refrigerator or you can save time by putting it in the freezer for one and a half hours.
Add the cooled mixture to your ice cream maker and follow the manufacturer's directions to freeze.
Scoop into a freezer-safe container and continue to freeze so that it hardens.