Line one 8-inch or 9-inch square pan with parchment paper, leaving an overhang on two sides. Alternatively, lightly grease the pan so the set fudge can be removed easily.

Add the sugar, butter, evaporated milk, pumpkin puree, and pumpkin pie spice to a heavy saucepan. Stir over medium heat until the butter melts and the mixture is evenly combined.

Bring the mixture to a full boil, stirring frequently. Once boiling, cook for about 5 to 8 minutes, stirring constantly, until a candy thermometer reads approximately 234 to 238°F (112 to 114°C), the soft-ball stage.

Remove the saucepan from the heat immediately. Quickly stir in the white chocolate chips, marshmallow creme, and vanilla extract until the mixture is completely smooth.

Pour the hot fudge into the prepared pan and spread it evenly with a spatula. Let it cool briefly at room temperature, then refrigerate for 2 to 3 hours, or until firm.

Lift the chilled fudge from the pan using the parchment overhang. Cut it into about 36 small squares with a sharp knife, wiping the blade between cuts for neat edges.