Preheat the oven to 350 degrees F (175 degrees C). Grease and flour two 10x15-inch jelly roll pans.

Beat eggs together in a bowl until combined; stir in 1/2 cup melted butter. Add cake mix, water, and pudding mix, stirring well to combine; the batter will be very thick. Divide batter between the prepared pans, spreading evenly.

Bake in the preheated oven until cakes spring back when pressed lightly with a finger or a tester comes out clean, 15 to 20 minutes. Cool completely on wire racks.

Beat confectioners' sugar, cream cheese, and 1/2 cup butter in a large bowl until smooth. Stir in whipped topping and vanilla extract until incorporated.

Spread filling on top 1 cooled cake layer; place remaining cooled cake layer on top and cut into bars. Wrap each bar in plastic wrap; store in the freezer.