Mix pumpkin puree, sugar, and pumpkin pie spice together in a large bowl. Slowly whisk in milk, then whipping cream, then vanilla extract. Refrigerate to allow flavors to blend, 2 hours to overnight.
Transfer pumpkin mixture into an ice cream maker; churn for 15 to 20 minutes. Freeze according to manufacturer's directions until it reaches 'soft-serve' consistency. Transfer ice cream to a lidded plastic container; cover surface with plastic wrap and seal. For best results, ice cream should 'bloom' in the freezer for at least 2 hours or overnight.