Preheat the oven to 350ºF (177ºC) and prepare three 6-inch or two 8-inch cake pans by spraying the sides with baking spray and placing a parchment paper circle into the bottom of each one.
In a medium bowl, whisk together the all-purpose flour, baking powder, baking soda, salt, cinnamon, allspice, nutmeg, and ground ginger. Set aside.
In a separate large bowl, add the brown sugar, granulated sugar, oil (if using melted butter in place of the oil, cool it for 5-10 minutes prior to adding here), eggs, pumpkin puree, and vanilla extract. Whisk together until uniform, then add the flour mixture and whisk until smooth.
Pour the batter evenly between the prepared cake pans and bake for 32-36 minutes, until a toothpick inserted comes out clean or with just a few moist crumbs. Cool the cakes in the pan for five minutes before removing and transferring to a wire rack to cool completely to room temperature.