Preheat oven to 350℉. Line three 8-inch cake rounds with parchment paper. Lightly grease and set aside.
In the bowl of a stand mixer fitted with a paddle attachment, beat butter until smooth. Add brown sugar and granulated sugar and mix with butter until smooth. Add eggs and vanilla. Mix on low speed until incorporated. Scrape down bowl as needed to ensure thorough mixing.
In a medium bowl, whisk together flour, baking powder, baking soda, and salt. Set aside.
In another bowl, mix together cocoa powder, espresso powder, and hot water. Mix until smooth. Allow to cool slightly.
Add 1/3 of flour mixture to mixing bowl. Mix on low speed until almost incorporated. Add sour cream and mix until combined. Add another 1/3 of flour mixture, followed with all of cocoa mixture. Mix on low speed. Add remaining 1/3 of flour mixture. Continue to mix on low speed until batter is smooth and thoroughly combined.
Divide batter among prepared cake pans. Use the back of a spoon or a mini offset spatula to level cake batter. Bake for 30-35 minutes until toothpick inserted in center of cake comes out clean. Rotate pans midway through baking. Allow to cool in pan for 5 minutes. Run a knife or offset spatula to loosen cake from pan. Unmold and allow to cool to room temperature on a wire rack.