Preheat the oven to 350 degrees F (175 degrees C). Line a 9x13-inch pan with enough parchment to have overhang on all sides.
Whisk together flour, pumpkin pie spice, salt, and baking soda in a bowl until well combined.
Whisk together pumpkin puree, white sugar, brown sugar, melted butter, egg, and vanilla in a large bowl until thoroughly combined. Mix in 1/2 of the flour mixture and stir until just combined. Stir in remaining flour mixture until just combined. Fold in white chocolate chips and walnuts. Pour batter into the prepared pan and spread into an even layer.
Bake in the preheated oven until the top springs back lightly when touched, about 45 minutes. Remove from the oven and let cool completely, 30 to 45 minutes. Slice into 24 bars.