Preheat oven to 350 degrees F with rack in the middle position. Generously butter three 8-inch round cake pans. (I only have 2 pans so I bake 2 layers first and then bake the 3rd layer.) Cut out rounds of parchment paper to fit the bottoms of the pans, place parchment rounds in the pans. Butter the parchment and then dust pans with flour, tapping out the excess.
In a medium bowl, whisk together the flour, cocoa powder, instant espresso powder, baking soda, and salt.
Place the milk and vanilla in a liquid measuring cup.
In the bowl of a stand mixer fitted with the paddle attachment, beat the butter on medium-high speed until smooth and shiny, about 30 seconds. With the mixer running, slowly add the sugar and beat until the mixture is pale, light, and fluffy, 3 to 5 minutes. Add the eggs 1 at a time, beating for 1 minute after each addition.
With the mixer on low speed, add about a third of the flour mixture, followed by a third of the milk mixture. Mix until barely incorporated and repeat with remaining flour and milk mixtures. Scrape the sides of the bowl and beat on low speed for 15 seconds more.
Divide the batter evenly between the three cake pans, spreading it out into even layers. Bake cakes until a tester comes out clean or with just a few crumbs attached, 23-26 minutes. Let cakes cool in pans on wire racks for 10 minutes.
Run a knife around the edge of each cake and invert pans onto racks; peel off parchment liners. Re-invert the cakes onto additional cooling racks. Cool cakes completely before frosting.