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. Tap out any extra flour.

In a medium bowl, whisk together the 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 remain.

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 speed to medium until light and fluffy, about 1 to 1 ½ minutes.

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

Add maple syrup and vanilla extract. Beat until the frosting is light and smooth, scraping the bowl as needed.

Frost the cooled cake evenly. Add chopped nuts on top if desired.