In a medium skillet over medium heat, melt the butter. Add the brown sugar and pumpkin pie spice. When the sugar has dissolved, add the spiced rum and cook for another minute (if cooking on a gas range, turn the flame off before adding the rum). Gently stir in the blueberries. Cook until the berries are heated through but still retain their shape, 3 to 4 minutes.
To serve, scoop the ice cream into bowls and top with the warm blueberry mixture.