Heat the oven to 350°F (175°C). Grease a 9x13-inch baking pan and line the bottom with parchment paper.
Whisk the flour, baking powder, baking soda, salt, cinnamon, ginger, nutmeg, cloves, and espresso powder in a large bowl until evenly combined.
In a second large bowl, whisk the granulated sugar, brown sugar, oil, and melted butter until smooth. Whisk in the eggs one at a time, then whisk in the pumpkin puree, milk, and vanilla until the mixture is smooth and glossy.
Add the dry ingredients to the wet in two batches, folding with a spatula just until no streaks of flour remain. Do not overmix — a few small lumps are fine.
Scrape the batter into the prepared pan and smooth the top. Bake at 350°F (175°C) for 32–35 minutes, until a toothpick inserted in the center comes out with just a few moist crumbs and the top springs back when gently pressed.
Cool the cake completely in the pan on a wire rack, about 1 hour. A warm cake melts frosting, so wait until the pan feels cool to the touch.
Beat the softened cream cheese and butter in a large bowl on medium speed until completely smooth and fluffy, about 2 minutes. Beat in the dissolved espresso, vanilla, and salt.
Add the powdered sugar in two batches, beating on low until combined, then on medium until the frosting is smooth and spreadable, about 1 minute. If it is too soft, chill it for 15 minutes before frosting.
Spread the frosting over the cooled cake in soft swoops with an offset spatula. Dust lightly with cinnamon and scatter a few chocolate-covered espresso beans over the top if you like.
Slice into 15 squares with a long, sharp knife, wiping the blade clean between cuts. Serve at cool room temperature or chilled.