Step 1: In a medium saucepan, combine the heavy cream, milk, and half of the granulated sugar over medium heat until it reaches a simmer.
Step 2: In a separate bowl, whisk the egg yolks with the remaining sugar until pale and slightly thickened.
Step 3: Slowly temper the hot cream mixture into the egg yolks, then return the entire mixture to the saucepan and cook until it coats the back of a spoon.
Step 4: Stir in the pumpkin puree, vanilla, pumpkin pie spice, and salt; strain through a fine-mesh sieve and chill the mixture for at least 6 hours.
Step 5: To make the swirl, melt the butter in a small pan, whisk in the brown sugar and cinnamon until dissolved, then let cool to room temperature.
Step 6: Churn the chilled pumpkin base in an ice cream maker according to manufacturer instructions.
Step 7: Layer the churned ice cream and the cinnamon mixture into a freezer-safe container, swirling gently with a knife, and freeze until firm.