Preheat the oven to 350ºF (177ºC).

Grease a 13 x 9-inch baking pan with cooking spray or vegetable oil, adding 1 tablespoon of flour to coat.

In a medium bowl, whisk together 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, then add pumpkin puree and whisk until combined.

Fold the flour mixture into the pumpkin mixture, stirring until just combined.

Spread the batter evenly in the prepared baking pan and bake until a toothpick comes out clean, about 30 to 33 minutes.

Cool the cake 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 combined, then increase speed and beat until light and fluffy.

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

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

Spread the cream cheese frosting over the cooled cake and sprinkle with chopped nuts if desired.