Preheat oven to 350°F (175°C). Line a 9x13-inch metal pan with parchment paper, leaving overhang on two sides, and lightly grease.
In a large bowl, whisk eggs until frothy. Add granulated sugar, brown sugar, and oil; whisk until smooth. Whisk in pumpkin puree and vanilla until no streaks remain.
In a separate bowl, whisk together flour, cinnamon, pumpkin pie spice, baking powder, baking soda, and salt. Break up any clumps.
Add dry ingredients to wet in two additions, folding gently with a spatula after each. Stop as soon as no dry streaks remain; batter should be thick but pourable.
Pour batter into prepared pan and spread evenly into corners. Bake for 28 to 32 minutes, until a toothpick in the center comes out clean or with a few moist crumbs.
Set pan on a wire rack and cool at room temperature for about 1 hour. Do not frost while warm.
Beat softened cream cheese and butter on medium speed until completely smooth. Add vanilla and salt; mix briefly. Add powdered sugar one cup at a time on low speed until thick and spreadable.
Spread frosting evenly over cooled cake with an offset spatula. Refrigerate at least 30 minutes to firm up, then lift out using parchment overhang and cut into 24 squares.