Set the oven rack to the middle position. Heat the oven to 350ºF (177ºC).

Grease a 13 x 9-inch baking pan with cooking spray or vegetable oil. Add 1 tablespoon of flour to the pan, turn, and tap the pan to lightly coat the bottom and sides.

In a medium bowl, whisk together 2 cups of sifted flour, pumpkin pie spice, baking powder, baking soda, and salt.

In a large bowl, whisk together granulated sugar, brown sugar, eggs, and vegetable oil until thick and smooth, about 3 minutes. Add pumpkin puree and whisk until combined.

Fold the flour mixture into the pumpkin mixture and stir until no flour streaks and just a few small lumps remain. Do not overmix.

Spread the pumpkin cake batter into the prepared baking pan. Bake until a toothpick in the center comes out clean, about 30 to 33 minutes. Cool completely on a wire rack for about 1 ½ to 2 hours.

In a stand mixer, beat softened butter and powdered sugar on low speed until just incorporated, then increase to medium speed until light and fluffy.

Add half of the softened cream cheese and beat on medium speed until incorporated. Add the remainder of the cream cheese and beat until smooth.

Add maple syrup and vanilla extract, and beat until the frosting is light and smooth.

Spread the cream cheese frosting over the cooled cake. Top with chopped nuts, if desired.