Preheat oven to 350 degrees. Grease and flour a 13x9-inch pan. For the cake, mix the ricotta, eggs, sugar, and vanilla; set aside. Prepare the cake mix according to package directions and pour it into the prepared pan. Immediately spread the ricotta mixture directly on top of the batter. Bake for 1 hour. Remove from the oven and let cool completely in the pan.
For the icing, stir the pudding mix into the milk, then fold in the Cool Whip. Spread the icing on the cake in the pan and refrigerate overnight.
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