Combine water, butter, sugar, salt & vanilla in a large saucepan. Heat until boiling & butter has melted.
Remove from heat & add flour, stir briskly with a spatula.
Using a hand mixer, quickly beat in eggs ONE at a time (so they don't scramble) beating well after each egg.
Spread dough into a lightly greased jelly roll pan. Bake at 400˚F for 25 minutes. The cake should puff up (it will be bumpy & uneven) & be lightly browned & crisp to the touch.
While cake is cooling, whisk pudding & milk. Let sit in the fridge until it firms up, about 5 minutes. Fold in ⅔ of the tub of whipped topping. Spread across the cooled cake.
Spread remaining whipped topping over the pudding layer. Drizzle with chocolate syrup. Cut & serve.