Place a rack in the center of the oven and preheat to 350 degrees. Butter and flour two 9" round cake pans and set aside.
In a medium bowl, whisk together the eggs, yolks, 1/3 cups buttermilk and the vanilla, set aside.
In the bowl of a stand mixer combine the flour, salt, baking soda and sugar. Blend for 30 seconds on low speed. Next add softened butter and remaining buttermilk. Once mixture is combined, mix for 2 minutes to fully incorporate. Add the egg mixture in 3 batches, scraping after each addition.
Divide the batter equally into 2 prepared cake pans. Bake for 25–30 minutes until a toothpick inserted into the center pulls out with just a few moist crumbs attached. Remove the cakes to a cooling rack and allow them to cool for 10–15 minutes before removing the cakes from the pans and cooling completely on the rack.
While the cakes are cooling, combine the whipped topping with the milk and vanilla. Add the instant pudding mix and beat on high speed for about 5 minutes until thick and creamy.
When the cakes have cooled, spread 1 cup of icing on only the top surface of the first cake. Place the second cake on top and cover the entire cake with the remaining icing. Cut the cake into large slices and enjoy!
Melted, rather than softened, butter can cause holes to form in your cake, known as ‘tunneling.’ It is best to set your butter out to allow it to reach room temperature on its own.