Position oven rack in the middle of the oven; preheat oven to 350°.
Butter an 8-inch square baking pan.
Whisk together the flour, 3/4 cup brown sugar, 2 tablespoons cocoa, the baking powder, and salt in a medium bowl; breaking up any large clumps of sugar with your hands.
Stir in the milk, butter, and vanilla with a wooden spoon just until blended; the batter will be very thick.
Transfer to the baking dish and smooth the top with a rubber spatula.
Stir together the remaining 1/2 cup brown sugar and 1/4 cup cocoa with a fork in a small bowl; breaking up any large clumps of sugar with your hands.
Stir in the nibs, if using; sprinkle the mixture evenly over the batter.
Pour the hot coffee evenly over the batter--do not stir.
Bake for 30-35 minutes, until the top layer is set and the dessert is beginning to pull away from the sides of the pan; the batter will have separated into cake and pudding layers.
Cool in the pan on a wire rack for at least 15 minutes.
Serve hot, warm, or at room temperature in small bowls, with whipped cream or ice cream, if desired.