In a small saucepan, set over medium heat, whisk together brown sugar and corn starch. Slowly pour in milk and whisk until smooth. Heat until steaming and beginning to thicken, whisking constantly.
In a small bowl, beat egg. Slowly pour in about half of the hot milk into the egg, whisking constantly until incorporated {this will help you not to make scrambled eggs when you add the egg to the hot mixture}. Pour the egg mixture back into the saucepan with the remaining milk mixture and whisk until thickened, about a minute or two.
Remove from heat and whisk in pumpkin puree, vanilla extract and pumpkin pie spice until well incorporated.
Transfer to a bowl. {Optional: Whenever I made pudding, custard, or anything with heated egg, I like to ensure there are no lumps by running it through a fine mesh strainer. Just pour the pudding in and press it through the strainer with a spatula or spoon to make sure that it's silky smooth.} Place a piece of plastic wrap directly on top of the mixture, patting it down lightly to make sure it's all covered. {This will prevent a "skin" from forming.}
Refrigerate until chilled.
Alternatively, you can make instant pumpkin pudding or use your favorite pudding recipe.