Position a rack in the center of the oven and heat the oven to 350°F Generously butter a 9-inch-square baking pan. Line the bottom of the pan with a square of parchment, butter the parchment, and then flour the bottom and sides of the pan. Tap out any excess flour.
If mixing by hand, put the softened butter and sugar in a medium bowl. Using a wooden spoon, cream them until smooth, about 1 minute. Switch to a whisk and blend in the eggs one at a time. Stir for another 30 seconds, until the batter is smooth and the sugar begins to dissolve. (If using a stand mixer, put the butter and sugar in the bowl and, using the paddle attachment, cream until smooth, about 1 minute. Blend in the eggs one at a time, mixing just until incorporated, about 20 seconds. Then switch to a whisk and blend in the rest of the ingredients by hand.) Mix in the vanilla and salt. Sift the flour, cocoa, baking soda, and baking powder directly onto the batter. Pour in the coffee. Gently whisk the ingredients until the mixture is smooth and mostly free of lumps.
Pour the batter into the prepared pan, spreading it evenly with a rubber spatula. Bake until a skewer inserted in the center comes out with only moist crumbs clinging to it, 40 to 43 minutes. Set the pan on a rack to cool for 20 minutes. Carefully run a knife around the edges of the pan, invert the cake onto the rack, and remove the pan. Invert again onto another rack and let cool right side up until just warm.
Make Ahead Tips: The cake can be wrapped in plastic and stored at room temperature for up to five days. Also freezes well.