In a medium saucepan, whisk together brown sugar and cornstarch until fully combined with no lumps remaining. You can use the back of a spoon to break up any lumps.
Gradually whisk in the unsweetened dairy-free milk until the mixture is smooth. Be sure to scrape the edges of the pan to incorporate any dry ingredients.
Add the pumpkin puree and whisk again until fully combined.
Place the saucepan over medium-high heat, and whisk continuously until the mixture thickens enough to coat the back of a spoon, about 5-7 minutes.
Once thickened, remove the saucepan from heat, and whisk in the pumpkin pie spice and vanilla extract.
Pour the pudding into a glass bowl and allow it to cool until warm.
Cover the surface of the pudding with plastic wrap to prevent a skin from forming, or strain it through a sieve for an ultra-smooth texture.
Refrigerate the pudding for at least 2 hours to set. Serve chilled, plain or with coconut whipped cream for added indulgence.