In a large saucepan, bring peanut butter, butter, and water just to a boil.
Immediately remove from the heat; stir in the flour, sugar, buttermilk, applesauce, eggs, baking powder, vanilla, salt, and baking soda until smooth.
Pour into a 15-by-10-by-1-inch baking pan coated with cooking spray.
Bake at 375 degrees for 15-20 minutes or until golden brown and a toothpick inserted near the center comes out clean. Cool on a wire rack for 20 minutes.
In a small saucepan, melt butter and peanut butter over medium heat; add milk. Bring to a boil. Remove from the heat.
Gradually whisk in confectioners' sugar and vanilla until smooth. Spread over warm cake.
Cool completely on a wire rack. Refrigerate leftovers.