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 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 mix until combined.

Fold the flour mixture into the pumpkin mixture until no flour streaks remain.

Spread the batter evenly into the prepared baking pan. Bake until a toothpick comes out clean, about 30 to 33 minutes. Cool completely on a wire rack.

In a stand mixer, beat softened butter and powdered sugar on low speed until just incorporated, then increase to medium and mix until light and fluffy, about 1 to 1 ½ minutes.

Add half of the softened cream cheese and mix until incorporated, then add the remainder and beat until smooth.

Mix in the maple syrup and vanilla extract until light and smooth.

Frost the cooled cake with the cream cheese frosting. Top with chopped nuts, if desired. Slice and serve.