Set the oven rack to the middle position and 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 to coat.

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

In a large bowl, whisk together the granulated sugar, brown sugar, eggs, and vegetable oil until thick and smooth, about 3 minutes. Add the 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 pan and bake until a toothpick comes out clean, about 30 to 33 minutes. Cool completely on a wire rack for 1 ½ to 2 hours.

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

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

Add the maple syrup and vanilla extract, beating until smooth.

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