In a medium saucepan, mix together the brown sugar and cornstarch. Add 1-2 tablespoons of the non-dairy milk, stirring until a thick paste forms.
Turn the stove to medium heat. Slowly pour in the remaining non-dairy milk, whisking constantly to prevent clumping. Stir in the pumpkin puree.
Bring the mixture to a boil, continuing to whisk. Lower to a simmer, stirring until thickened to pudding consistency. This will take 3-4 minutes.
Stir in the cinnamon, nutmeg, allspice, vanilla extract, and maple syrup.
Place your 9" pie dough in a pie dish. If you're using the oil pie crust, I recommend avoiding any fancy edges. It'll be easiest if you just make a flat, simple edge.
Pour the pumpkin pudding mixture into the unbaked pie crust. Smooth into an even layer with a spatula. Place the pie on the middle rack of the oven. Bake at 375F for 45-50 minutes. See note for additional baking tips.
The pie is cooked when it is slightly jiggly in the center, but not liquidy. Remove from the oven and let cool to room temperature. After cool, refrigerate for 3-4 hours, until chilled.
Once chilled, slice and enjoy with non-dairy whipped topping and a drizzle of maple syrup.
This recipe was tested using a few different pie dishes. The pie takes longer to bake and set depending on the thickness of the dish you use. The time on this recipe is based on a thin metal pie dish, the disposable kind. Add 5 minutes to the baking time if using a thicker ceramic or glass dish.