Step 1

Adjust oven rack to middle position and heat oven to 325 degrees. Melt butter, chocolate, and 1/3 cup cocoa together in 10-inch cast-iron skillet over low heat, stirring often, until smooth, 2 to 4 minutes. Set aside to cool slightly.

Step 2

Whisk flour, baking powder, and salt together in bowl. In separate bowl, whisk brown sugar, 1/3 cup granulated sugar, and remaining 1/3 cup cocoa together, breaking up any large clumps of brown sugar with fingers. In third bowl, combine coffee and water.

Step 3

Whisk milk, vanilla, egg yolk, and remaining 2/3 cup granulated sugar into cooled chocolate mixture. Whisk in flour mixture until just combined. Sprinkle brown sugar mixture evenly over top, covering entire surface of batter. Pour coffee mixture gently over brown sugar mixture.

Step 4

Transfer skillet to oven and bake until cake is puffed and bubbling and just beginning to pull away from sides of skillet, about 35 minutes, rotating skillet halfway through baking. Using potholders, transfer skillet to wire rack and let cake cool for 15 minutes. Serve.