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For the Filling:
Make the pudding according to the package directions and chill. The pudding may be used immediately or chilled and covered up to one day before assembling the cake. Thaw the whipped topping in the refrigerator for 4 hours before assembling the cake.
For the Chocolate Sauce:
In a 1-quart saucepan over medium high heat, combine the milk, butter, and salt. Bring the mixture to a boil.
Combine the sugar and cocoa powder and stir to avoid any lumps, then add to the pan and continue to stir with a whisk until it boils and is thickened, at least 2 minutes. The mixture should coat the back of a spoon when thickened. It will continue to thicken as it cools.
Transfer the chocolate sauce to a small bowl and cover. Chill for 1 hour before using. The sauce can be stored in the refrigerator up to a week. Bring the sauce to room temperature and stir before using.
For the Dough:
Combine the water and the butter in a 4-quart heavy saucepan and bring it to a simmer. Once the butter has completely melted, add the flour and stir well. Continue to stir over medium heat for 2 minutes. It is important to cook it the full 2 minutes so that the flour will cook completely. The dough will stick to the sides of the pan and then form a ball. Keep stirring and cooking.
After 2 minutes, remove the pan from the heat and place the dough in the bowl of an electric mixer. Allow the dough to cool for about 5 minutes.
With the machine running on low, add the salt and one egg at a time. Allow each egg to incorporate before adding the next. The dough will change in appearance from looking like scrambled eggs to looking like shiny batter. Continue to beat for about 2 minutes until the dough pulls away from the bottom or sides of the bowl.
Spread the dough into a greased 9 x 13-inch baking dish, pushing the dough into the corners and up the sides.
Preheat oven to 425 degrees F and bake for for 15 to 18 minutes. The cake should look lightly golden brown. Remove from the oven and cool.
Assembling the Cake:
After the base has cooled completely, spoon the prepared pudding into the center, smoothing evenly and pushing to the edges. Top with the whipped topping, smoothing evenly and swirling in a decorative pattern. Drizzle with about 1/3 cup of the chocolate sauce. Cover cake lightly with plastic wrap and chill for 1 hour before serving.