4 tablespoons unsalted butter (1/2 stick), melted and cooled slightly
1 large egg
1 tablespoon vanilla extract
3/4 cup packed dark brown sugar
3/4 cup strong brewed coffee, at room temperature
Heat the oven to 350°F and arrange a rack in the middle. Coat an 8-by-8-inch square baking dish with butter and set aside.
Combine flour, 3/4 cup of the cocoa, baking powder, and salt in a medium bowl and whisk to break up any lumps. In a large bowl, whisk together 1/2 cup of the granulated sugar, milk, butter, egg, and vanilla until evenly combined and smooth. Add flour mixture to milk mixture and whisk until just combined (some lumps will remain).
Transfer batter to the prepared baking dish and spread evenly. Combine remaining 1/4 cup cocoa, remaining 1/4 cup granulated sugar, and brown sugar in a small bowl until well mixed, then sprinkle over batter. Pour coffee over batter, do not stir, and place in the oven.
Bake until cake is bubbling, puffed, and set at the edges but still a bit loose in the middle, about 25 minutes. Remove to a rack and let cool 5 minutes before serving.