Preheat the oven to 350°F (175°C). Prepare three 8-inch cake pans by greasing them and lightly flouring, tapping out any excess flour, or using a commercial cake release.
In a large bowl with a stand mixer or using a hand mixer, cream the softened butter and sugar on medium-high speed until light and fluffy, about 3 minutes.
Add the eggs one at a time, mixing until each is fully incorporated before adding the next.
In a separate bowl, sift together the all-purpose flour, cocoa powder, kosher salt, and espresso powder; set aside.
Combine the buttermilk and vanilla extract in a measuring cup and set aside.
With the mixer on low, add the dry flour mixture and the buttermilk mixture to the butter mixture alternately, beginning and ending with the flour mixture. Mix until just combined and scrape down the bowl as needed.
Reduce the mixer speed to low, stir in the baking soda until incorporated, then add the vinegar and mix briefly to combine without overmixing.
Divide the batter evenly among the prepared pans and smooth the tops. Bake for about 25 minutes, or until the cakes pull away slightly from the pan edges and a toothpick inserted in the center comes out clean.
Remove the cakes from the oven and let them cool in the pans for about 10 minutes, then turn them out onto cooling racks and cool completely before frosting.