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, vigorously whisk together granulated sugar, brown sugar, eggs, and vegetable oil until thick and smooth, about 3 minutes. Add pumpkin puree and whisk until combined, about 15 seconds.

Fold the flour mixture into the pumpkin mixture and stir until no flour streaks and just a few small lumps remain.

Evenly 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 fitted with a paddle, beat softened butter and powdered sugar on low speed until just incorporated. Gradually increase speed to medium and beat until light and fluffy, about 1 to 1 ½ minutes.

Add half of the softened cream cheese and beat on medium speed until incorporated, about 15 seconds. Add the rest of the cream cheese and beat until smooth, about 15 to 20 seconds.

Add the maple syrup and vanilla extract. Beat until the frosting is light and smooth, scraping the bowl as needed, about 30 to 45 seconds.

Frost the cooled cake and sprinkle with the chopped nuts if desired.