In an electric mixer, cream together cane sugar, coconut sugar, and room temperature butter until smooth.
Add vanilla extract and milk, then whisk again.
Mix in all-purpose flour, baking powder, cinnamon, 1 tsp pumpkin pie spice, and salt until fully combined.
Divide the dough in half. Mix pumpkin puree and remaining pumpkin pie spice into one half.
Roll out plain dough between parchment paper sheets. Spoon pumpkin dough on top, spread evenly, then roll tightly into a log. Chill for 30 minutes.
Preheat oven, slice the log into 1/2-inch pieces, and place on a parchment-lined baking sheet.
Bake for 12-15 minutes, then let cool for a couple of minutes before enjoying.