Preheat oven to 350°F. Spray a 10-inch oven-safe skillet with baking spray. Set aside.
In a medium bowl, whisk together the flour, sugar, cocoa powder, baking powder, and salt.
In a separate small bowl, mix the milk, butter, and vanilla until well combined.
Add the wet ingredients to the dry ingredients and stir until combined. (It will be thick.)
Stir in the chocolate chips. Spread the batter evenly into the prepared skillet.
In a small bowl, mix the brown sugar and cocoa powder. Sprinkle this mixture evenly over the batter in the skillet.
Carefully pour the hot coffee over the top of the batter and brown sugar mixture. Do not stir.
Bake for 30-35 minutes, or until the cake is mostly set. The edges may still be slightly jiggly, but the cake will be firm to the touch on top.
Remove the cake from the oven and let it cool for 5-10 minutes. During this time, the liquid will thicken into a rich pudding beneath the cake.
Serve warm, scooping the cake and pudding together into bowls. Optionally, top with whipped cream or vanilla ice cream.