In a medium sauce pot, combine cream, milk, and one tablespoon brown sugar. Set over medium heat and bring to a simmer.
Meanwhile, in a large bowl, whisk together egg yolks, remaining brown sugar, cinnamon, pumpkin pie spice mix, and salt. Once milk mixture has almost come to a boil, lower heat. Temper hot milk into sugar-yolk. Slowly add hot milk into sugar-yolk mixture about ¼ cup at a time, whisking immediately to distribute heat. Continue adding hot milk to yolk mixture until there are only a couple tablespoons of milk left in the pot.
Transfer the tempered egg mixture back into the sauce pot. Stir over medium-low heat until mixture reaches nappe consistency, or 180°F on a kitchen thermometer. Be sure to scrape along the sides of the pot to ensure mixture does burn along edges. Turn off heat and remove pot from heat source.
Stir in pumpkin puree and bourbon until smooth. Transfer mixture to a heat-safe container. Place plastic wrap directly onto ice cream base. Cover container and keep in the fridge until cold. Ideally, chill overnight.
Once mixture is cold, churn ice cream base according to your ice cream maker machine's instructions.
Churn until the volume of the ice cream mixture has increased by half and has thickened to a soft serve consistency. Transfer ice cream to a freezer safe container. Press a piece of plastic wrap on the surface of the ice cream, cover container, and keep in the coldest part of the freezer for at least one hour to firm up.