Prepare 1 box yellow cake mix according to the package directions, using the eggs, vegetable oil, and water called for on the box.
Pour the batter into a well-greased 9x13-inch baking pan and bake according to the package directions.
Remove the cake from the oven and, while it is still warm, use the handle of a wooden spoon to poke holes about 1 inch apart all over the cake, making sure the holes reach the bottom.
In a large mixing bowl, whisk together 2 boxes (3.4 ounces each) instant vanilla or French vanilla pudding mix with 4 cups milk for about 1 minute, or until most of the lumps are gone.
Immediately pour the pudding evenly over the warm cake, directing as much pudding as possible into the holes.
Use the back of a spoon or an offset spatula to gently spread the pudding and press it into the holes.
Refrigerate the cake for about 2 hours, or until the pudding is set and the cake is completely cooled.
Remove the lid and foil from 1 tub chocolate frosting and microwave it for 10 to 15 seconds until pourable but not hot. Stir until smooth.
Pour the frosting over the chilled pudding layer and spread it evenly from the center to the edges.
Let the cake sit at room temperature for a few minutes, then refrigerate until the frosting is fully set before slicing and serving.