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Heat the oven to 350°F with a rack in the middle position. Mist an 8-inch square glass baking dish or metal baking pan with cooking spray.
To make the cake, in a large bowl, whisk together the flour, cocoa, brown sugar, baking soda and salt, breaking up any lumps of cocoa and sugar. Make a well in the center and to it add the coffee, oil, vinegar and vanilla; whisk the wet to combine, then whisk them into the dry ingredients just until evenly moistened. Do not overmix. The batter will be fluid.
Pour the batter into the prepared baking dish. Bake until a toothpick inserted at the center of the cake comes out clean, 50 to 55 minutes. Cool completely in the pan on a wire rack.
When the cake has cooled, make the icing. In a large bowl, whisk the sugar and cocoa until no lumps remain. Add the coffee, vanilla and oil; whisk until smooth. The icing should be thick but spreadable, with a consistency similar to pourable yogurt. If it is too stiff, whisk in additional coffee 1 teaspoon at a time until the proper consistency is achieved. Pour the icing onto the center of the cake in the pan, then spread evenly. Let stand at least 30 minutes before cutting and serving.