Bake cake according to directions on box and pour into a 9-by-13-inch pan. Cook until toothpick inserted into cake comes out clean. Let cake cool for about 5 minutes.
While cake is cooling, make the nougat filling. Melt butter in a saucepan over medium heat. Add sugar and milk, stirring constantly until sugar is dissolved. Bring to a boil and stir for 4 minutes. Remove from heat and stir in marshmallow cream and vanilla. Mix until smooth.
Using the handle end of a wooden spoon, poke 10 to 15 holes in the cooled cake.
Evenly pour and distribute nougat filling into the holes in the cake. Next, evenly pour and distribute the caramel topping into the holes in the cake.
Let cake cool completely. Once cake is cooled completely, evenly spread thawed Cool Whip over the top of the cake. Sprinkle chopped Milky Way bars over the top of the Cool Whip. Store in refrigerator until ready to serve.