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 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, about 15 seconds.

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

Evenly spread the pumpkin cake batter into the prepared baking pan. 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 softened butter and powdered sugar on low until incorporated. Increase to medium speed and beat until light and fluffy, about 1 to 1 ½ minutes.

Add half the cream cheese and beat on medium until incorporated, about 15 seconds. Add the remainder and beat until smooth, about 15 to 20 seconds.

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

Frost the cake as desired. Sprinkle chopped nuts on top if using. Slice and serve.