Preheat the oven to 350 degrees F. Butter an 8x8-inch baking dish and set aside.
In a medium mixing bowl, whisk together the flour, ⅔ cup sugar, ¼ cup cocoa powder, baking powder, and salt until combined. In a separate bowl, whisk together milk, butter, and vanilla until smooth. Add the wet mixture to the dry mixture and stir to combine into a thick batter. Pour into the prepared baking dish.
In a small bowl, stir together the remaining sugar and cocoa powder. Sprinkle evenly over the batter, then sprinkle on chocolate chips. Slowly pour the boiling water over the top of the mixture without stirring.
Bake for 30-40 minutes, until the center is just set. Serve slightly warm with ice cream.
Bake this chocolate cobbler with a sheet pan underneath your baking dish. This will help to catch any drips so they don't end up in the bottom of your oven.
Let your chocolate cobbler cool for 5 minutes before serving.