Brat Chicken is interesting version of the mud chicken story.
One other popular version is the "Beggars Chicken", as in beggars got/caught a chicken and try to cook it without tools. The beggar wrapped it in mud and the threw it into fire.
its also known as Prosperity Chicken as the myth have it, a royal/emperor saw the beggar cook it that way, got curious and like the result. The royal/emperor then give the said beggar significant sum of wealth for the discovery. Hence the alternate name.