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 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 smooth. Add pumpkin puree and whisk until combined.

Fold the flour mixture into the pumpkin mixture until no flour streaks are visible.

Spread the batter into the prepared pan and 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 the softened butter and powdered sugar until just incorporated.

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

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

Add the maple syrup and vanilla extract, and beat until the frosting is light and smooth.

Spread the cream cheese frosting evenly over the cooled cake.

Sprinkle with chopped nuts on top if desired. Slice and serve.