Place an oven shelf in the center position and preheat the oven to 350 degrees F. Grease a 9x13-inch nonstick pan with baking spray or butter.
In a medium-sized bowl, whisk together the flour, baking powder, baking soda, salt, cinnamon, allspice, nutmeg, and ginger.
In a stand mixer fitted with the paddle attachment (or in a large bowl with an electric hand mixer, see notes), cream butter and sugar for several minutes until light and fluffy. Turn the speed down to low and add the eggs, one at a time, making sure each egg is fully incorporated before adding the next. Scrape down the bowl a few times with a spatula during this process. With the mixer on low speed, afternate between adding the dry ingredients and the pumpkin puree, starting and ending with the dry ingredients, mixing until the batter is smooth.
Spread the batter into the prepared pan, using a regular or offset spatula to level out the top. Bake for 30-35 minutes, until a toothpick inserted into the center comes out clean. The top will look slightly darker in color, and should feel firm but spring back when lightly pressed. Remove from the oven and allow to cool.
Prepare the frosting
In a stand mixer fitted with the paddle attachment, or using a large bowl and an electric hand mixer, combine the cream cheese, butter, and vanilla extract on medium-high to high speed until smooth.
Turn the speed down to low and add the salt along with the sugar in 3-4 batches, then turn the speed back up to high and mix until the frosting is smooth.
Once the cake has cooled to room temperature, frost, slice and serve. Store leftovers in an airtight container in the refrigerator for up to one week.
Notes
I found the batter to be smoother with an electric hand mixer vs a stand mixer, but it took longer to cream the butter and sugar. The first time I prepared this cake, I used a 9x13 Pyrex dish, and it did not cook as evenly as when I used a 9x13 nonstick pan. With the Pyrex dish, the edges browned too much before the center of the cake was fully baked. I recommend preparing this cake in a nonstick pan such as this one.