Preheat the oven to 350°F. Line a 9x13 inch pan (for classic bars) or a 9x9 inch pan (for thick squares) with parchment paper, leaving a little overhang on two sides, and spray it lightly with nonstick cooking spray.
In a medium bowl, whisk together the flour, baking powder, baking soda, salt, cinnamon, ginger and cloves. Set aside.
In the bowl of a stand mixer, or in a large bowl with a hand mixer, beat the eggs and sugar on medium speed for 1 to 2 minutes, until pale and slightly thickened. Beat in the oil, vanilla and pumpkin until smooth.
Add the flour mixture to the pumpkin mixture and mix on low speed, or stir by hand with a spatula, just until no dry flour remains. The batter will be thick. Stop mixing as soon as it comes together.
Spread the batter evenly in the pan. Bake until the center springs back when lightly pressed and a toothpick in the middle comes out clean: 25 to 30 minutes in a 9x13 inch pan, or 35 to 45 minutes in a 9x9 inch pan.
Cool the bars completely in the pan on a wire rack, about 1 hour. Warm bars will melt the frosting.
For the frosting, beat the cream cheese and butter with an electric mixer until smooth. Add the powdered sugar and mix on low speed until combined, then beat in the vanilla. Add milk a teaspoon at a time only if the frosting is too stiff to spread.
Lift the cooled slab out of the pan with the parchment and set it on a cutting board. Spread the frosting over the top and cut into 24 bars (9x13) or 16 squares (9x9). Store covered in the refrigerator.