Adjust oven rack to middle position and preheat oven to 350 degrees. Prepare two 8-inch or 9-inch round cake pans with non-stick spray. Line the bottoms with parchment paper, and spray again with non-stick spray.
In a large bowl, whisk together the flour, cocoa, instant espresso, baking soda, and salt. Set aside. In the bowl of a stand mixer fit with the whisk attachment on medium speed, beat together the butter and sugar until light and fluffy, about 3 to 6 minutes. Beat in the eggs one at a time into the mixture until incorporated, scraping down the bowl as necessary. Beat in the vanilla.
With the mixer speed reduced to low, beat in one-third of the flour mixture, then half of the milk. Beat in half of the remaining flour, then the remaining milk. Beat in the remaining flour mixture. Using a rubber spatula, give the batter a final stir to ensure the mixture is thoroughly combined.
Divide the batter evenly between both pans, smoothing the tops. Bake until a toothpick inserted in the center of the cake comes out dry with a few crumbs attached, 35 to 40 minutes. Rotate the pans halfway through baking time.
Cool the cakes in the pans for 10 minutes. Run a small sharp knife around the inside of each pan to loosen, then flip the cakes onto a wire rack. Peel off the parchment paper then flip the cakes right side up to cool completely, about 1 to 2 hours. Cool the cakes completely before frosting.