Preheat oven to 350F (177C) and place a rack in the center of the oven.
In the bowl of an electric stand mixer fitted with the paddle attachment, add the butter, sugar and finely ground espresso and mix on medium speed until light and fluffy, about 4-5 minutes. With the mixer on low, stream in the milk. Scrape the sides and bottom of the bowl to make sure everything is well blended. Add in the sour cream and mix for 1 minute more until light, fluffy and fabulous. With the mixer on low, add in the eggs, one at a time, making sure that each is well blended before adding in the next. Add in the vanilla and run the machine for 30 seconds more. Take the bowl out of the mixer.
In a medium bowl whisk together the flour, cocoa powder, baking powder, baking soda and salt and whisk to blend completely. Fold the flour mixture into the butter mixture in three batches, mixing each until almost (but not quite) blended. Finally, pour in the hot espresso and stir. Scrape the sides and bottom of the bowl to make sure everything is well blended. Stir the batter enough that everything is combined but being cautious not to mix too much as we don't want a tough cake. Set aside.
Grease your 9 x 13 baking dish (23 x 33–cm) and line with parchment paper, taking care that it lays flat against the dish and letting the excess hang over the sides of the pan (you'll use these to pull the cake out of the pan). Pour the batter into the prepared baking dish. Tap on the counter several times to settle the cake batter and release any trapped bubbles. Use a spatula to smooth and even out the top of the cake batter. Bake in the center of the oven for about 40-43 minutes. Use a toothpick or cake tester to test for doneness. Some moisture and crumbs are fine, but if it's coated in cake batter, bake it for several minutes more. Another way to test for doneness: press the center part of the cake gently and it should spring back when your finger lifts.