A Brown Joe, is a sexual act, where feces and fellatio becomes one.
Person A proceeds to defecate in Person B's mouth, and while his droppings are still hot, Person A begins a fellatio with Person B. The term is a play of words, where the "Brown Joe" is the penis of person B when eventually rectracted. Can also be read as "Brown Dough", a term for the defecation process preceeding the act.
Ali and I did a Brown Joe yesterday. I was on the recieving end. The recieving end of the turd, you know.