Chicken Egg

Chicken Egg summons and dismisses Westfall Chicken, a chicken companion that follows you around, meaning it is a non-combat pet and will not aid you in battle.

Source
Unlike most vanity pets, it is gained in a highly unusual method.

Every chicken you see in the game has a small chance of offering you a quest when you use the /chicken (or /flap) emote in front of it. If you cluck at a chicken enough, it will "looks at you puzzled. Perhaps you should inspect it". The chicken will change from to, and offer a quest. Talk to it to get CLUCK!. Note that the exclamation mark may not show up above the chicken unless you are very low level. The quest can be accepted by anyone nearby. The quest will only be offered for a short time, however.

The quest requires you to bring Special Chicken Feed to the chicken. The feed can be purchased from in Westfall or  in Brill. Once the quest is completed, the chicken will lay a Chicken Egg which you can take to learn how to summon a chicken.

Note that chickens can be found within a few yards of Farmer Saldean in Westfall. In Brill, a chicken may be found just inside the gate of graveyard to the west.

For your own sanity, it is strongly suggested you write a macro that does /chicken (or /flap) over and over to activate this quest.

Pet Journal
Don't call this bird chicken unless you want egg on your face.

Macro
This macro will do the chicken emote 50 times: /run for i = 1, 50 do DoEmote("CHICKEN") end

Another method would be to type "/chicken" into Notepad or any writing program available, then copy/paste exponentially (ie. copy then paste five times, then copy those five and paste two times, copy those 15, etc.):

/chicken /chicken /chicken

etc.

Once you have a large batch, copy, open up WoW, make a new macro and Ctrl-V into the Command window. In reality though, you can only fit 28 complete lines into the window, taking up 251 of the 255 characters available. You will only have to use the macro twice, if even that, to trigger the quest.

You can also create a macro that has just 1 /chicken in it and put on your action bar. Then just sit there and spam it till the chicken pops a quest.