Preheat your oven to 350 degrees Fahrenheit. Grease a 15 by 10 inch jelly-roll pan generously with butter or nonstick baking spray. This cake is thin, so a proper coating prevents sticking.

In a large mixing bowl, whisk together the flour, granulated sugar, baking soda, and salt. Set the dry ingredients aside while you prepare the wet mixture.

In a medium saucepan, combine the butter, water, and creamy peanut butter. Place the pan over medium heat and stir occasionally. Bring the mixture to a rolling boil. Once boiling, immediately remove the saucepan from the heat.

Pour the hot peanut butter mixture over the dry ingredients. Whisk or stir with a wooden spoon until completely combined. The mixture will be thick at first but will loosen as you mix.

In a separate small bowl, whisk the beaten eggs, vanilla extract, and buttermilk together. Pour this into the peanut butter batter and stir until the batter is smooth and glossy. Be sure to scrape the bottom and sides of the bowl.

Spread the batter evenly in the prepared jelly-roll pan. The layer of batter will be thin, so use an offset spatula to reach the corners.

Bake for 25 to 28 minutes, or until the cake springs back when gently touched in the center. A toothpick inserted in the middle should come out clean or with a few moist crumbs.

While the cake bakes, prepare the icing. In a clean saucepan, combine the butter, peanut butter, and milk. Warm over medium heat until the butter melts and the mixture reaches a boil. Stir frequently to prevent scorching.

Remove the saucepan from the heat and whisk in the powdered sugar one cup at a time. Add the vanilla extract. Keep whisking until the icing is smooth and glossy. It should have a pourable but spreadable consistency.

As soon as the cake comes out of the oven, pour the warm icing over the top. Use an offset spatula to spread it quickly from edge to edge. The icing will firm up as it cools. Let the cake rest for at least thirty minutes before slicing.