Preheat your oven to 350°F (175°C). Lightly grease a 9×13-inch baking pan with cooking spray or butter. For even easier removal, you can line the pan with parchment paper, leaving an overhang on the sides to lift the bars out later.
In a medium-sized mixing bowl, whisk together the all-purpose flour, baking soda, baking powder, salt, cinnamon, cloves, nutmeg, and ginger. Ensuring these are well-combined prevents any clumps of spices or leavening agents in your bars.
In a large bowl, beat the four eggs with the granulated sugar and brown sugar until the mixture is smooth. Pour in the vegetable oil and the pumpkin puree, whisking until everything is fully incorporated and the color is a vibrant orange.
Gradually add the dry ingredient mixture into the wet ingredients. Use a spatula or a whisk to fold them together gently. Stir only until no streaks of flour remain; overmixing can lead to a tough texture rather than a chewy one.
Spread the batter evenly into your prepared 9×13 pan. Place it in the center of the oven and bake for 25 to 30 minutes. You can check for doneness by inserting a toothpick into the center; it should come out clean or with just a few moist crumbs attached.
Remove the pan from the oven and place it on a wire rack. It is vital to let the bars cool completely before frosting. If the bars are even slightly warm, the cream cheese frosting will melt and slide off.
While the bars cool, beat the softened cream cheese and butter together in a bowl until smooth and creamy. Slowly add the powdered sugar and vanilla extract. Beat on high speed for 2-3 minutes until the frosting is light and fluffy.
Once the pumpkin base is cold, spread the frosting over the top in an even layer using an offset spatula. Slice into 24 squares and serve.