Heat oven to 350°F (175°C) and grease an 8×8-inch square pan. In a large bowl, whisk the flour, sugar, cocoa, instant coffee, baking powder, and salt until combined.

Make a well in the center, add the egg, milk, oil, and vanilla, and stir just until no dry flour remains.

Fold the batter a few times with a rubber spatula to catch any pockets, then scrape it into the prepared pan and smooth the top.

Bake for 40 to 45 minutes, until a toothpick inserted in the center comes out with moist crumbs. Cool in the pan 10 minutes, then transfer to a rack to cool completely.

Whisk the powdered sugar, instant coffee, and 2 tbsp milk until smooth, adding the remaining 1 tbsp milk if needed for a pourable consistency. Drizzle over the fully cooled cake and let set 15 minutes before slicing.