The Bush Chicken is a possible drop from the new reward bag, Pilgrim's Bounty, from the holiday dailies. The five dailies aren't account-wide so you can complete them on multiple characters. All you need is cooking, any skill level works. You can do the quests even with cooking 1; as you complete quests, you'll be required to cook different/more things which in turn raises your cooking skill.
I had to level cooking on a few of my toons, which was time consuming but worth it in the end. The only issue was the RNG. There were reports that some had the Bush Chicken drop on their first few toons, while others said they didn't get the pet until 8+ toons down the line.
In the end I was part of the latter camp, and after nothing on the first day (30+ bags, that's 6+ toons) I finally received my little chicken on the second day. :D
I won't go into any kind of guide or walk through, but here are some things that I observed while working my way through the dailies.
• You can level cooking to 300 with JUST the holiday food that's required for the dailies.This might seem like a grind, with a relatively low drop rate, AND it's limited time (only one week) but the consolation is that it can be completed on multiple toons each day. As long as your character can learn cooking 300 (level restrictions may apply), you can have an army of toons farming this pet for you. It WILL eat up a large portion of your day, though, but little things like the Chef's Hat and mage portals help.
• While leveling cooking using the holiday recipes, don't vendor the food. You can save them and turn them in another day.
• Purchase extra materials from each city's vendor (mainly city specific materials; pumpkins, cranberries, sweet potatoes), cook them all at once, and save them to turn in the next day. The turkeys are a bit of a pain though, but easy enough to find/kill.
• Mages are a godsend! My OH and I teamed up and we made use of each other's mages heh.
• Chef's Hat makes things go by much faster. Especially if you're leveling cooking and need to make 100 pumpkin pies or whatever.
Overall, I'm satisfied. I put time and effort into collecting a pet and was eventually rewarded. Plus now I have a ton of toons with at least 300 cooking, if not higher! :P