In a medium mixing bowl, whisk together the pumpkin puree, milk, heavy cream, granulated sugar, brown sugar, eggs, and vanilla extract until well combined.
In a separate small bowl, mix together the ground cinnamon, ground nutmeg, salt, and cornstarch.
Gradually add the spice mixture to the pumpkin mixture, whisking thoroughly until everything is evenly blended.
Pour the entire mixture into a 2-quart saucepan. Cook over medium heat, stirring constantly for about 20 minutes, or until the mixture thickens and starts to bubble.
Once thickened, remove from heat and allow the pudding to cool slightly for about 5 minutes.
Pour the pudding into individual serving dishes or cups. Cover with plastic wrap or a lid to prevent a skin from forming.
Chill in the refrigerator for at least 2 hours before serving to allow the flavors to meld and the pudding to set properly.
Serve chilled, and you can garnish with whipped cream and a sprinkle of cinnamon if desired.