Preheat and Melt: Adjust oven rack to the middle position and preheat oven to 325°F. In an 8-inch oven-safe skillet, melt butter, chocolate, and 1 tablespoon cocoa powder over low heat, stirring frequently until smooth (about 2 minutes). Remove skillet from heat and let cool slightly.
Prepare Cocoa-Sugar Mixture: In a bowl, whisk together 1 tablespoon granulated sugar, brown sugar, and the remaining 2 tablespoons cocoa powder. Break up any clumps of brown sugar with your fingers.
Mix Batter: Whisk together milk, egg yolk, vanilla, baking powder, salt, and the remaining ¼ cup granulated sugar into the cooled chocolate mixture. Slowly whisk in flour until just combined. Sprinkle the prepared brown sugar mixture evenly over the batter, covering the entire surface. Pour boiling water over the brown sugar layer.
Bake: Scrape down the exposed sides of the skillet and transfer to the oven. Bake uncovered for about 25 minutes, rotating the skillet halfway through. The cake is done when it begins to pull away from the sides of the skillet and the top is just firm to the touch.
Cool and Serve: Remove the skillet from the oven and transfer it to a wire rack. Let cool for 10 minutes before serving. Enjoy warm with whipped cream.