Preheat your oven to 350°F (175°C). Line a 9x13-inch baking pan with parchment paper. Generously spray the sides and the bottom of the pan with nonstick cooking spray, then set it aside.
In a large mixing bowl, combine granulated sugar, light brown sugar, all-purpose flour, Dutch-process cocoa powder, baking soda, and salt. Mix on low speed until all the dry ingredients are thoroughly combined. If you find any large clumps, don’t hesitate to use your hands to break them up!
In another bowl, mix the eggs, sour cream, whole milk, vegetable oil, and vanilla extract until completely incorporated.
Pour the wet mixture into your dry ingredients. Stir gently until just combined. It’s okay if the batter is thick; just ensure not to over mix. Overmixing can lead to a dense cake.
In a large measuring cup, combine hot water and espresso powder. This mixture will further enhance the chocolate flavor. Allow it to sit for a minute before gently whisking until completely smooth.
Pour the hot water mixture into the cake batter and gently fold it in. You’ll notice that the batter becomes quite thin; this is normal, so don’t worry!
Pour the batter into the prepared pan and bake in your preheated oven for about 35 minutes. To check for doneness, insert a wooden toothpick or cake tester in the center. It should come out clean or with just a few moist crumbs.
Once baked, let the cake cool in the pan for about 10 minutes. After that, invert the cake onto a cooling rack to cool completely. This step is crucial for allowing the cake to firm up before frosting.
Now, for the frosting! In a bowl of a stand mixer fitted with a paddle attachment or a large bowl with a handheld mixer, beat the unsalted butter on medium speed until it’s completely smooth and creamy.
Reduce the speed to low and gradually add the confectioners' sugar, mixing until well combined. The frosting should be smooth and creamy at this point.
Add in the salt and milk, beating until smooth and fluffy.
In another bowl, mix instant espresso powder with the melted white chocolate until smooth. Scrape this mixture into the frosting and beat everything together until well incorporated.
Once all ingredients are well-mixed, increase the speed to medium-high and beat for about one minute until fluffy.
Spread the frosting generously over the top and sides of the cooled cake. If you're feeling fancy, sprinkle it with chocolate shavings for that extra touch.
Allow the frosting to set for about 20 minutes before slicing and serving. Enjoy your delicious creation!