If you go to the Ganache Mine District (on any day of the week except for Saturday) and head to where Dale and the other carpenters are, you can get a Chicken Coop built for 2,400G. You will also need 10 material stone and 10 lumber. It should take 2-3 days for the Chicken Coop to be completed.
If you buy a mayonnaise maker, it will be at the back of the coop where you can make mayonnaise just by putting in an egg from either a chicken, a duck or an ostrich.
For more gold, material stone, lumber, and a few more days, you can upgrade your Chicken Coop. Upgrades in the Chicken Coop will increase the overall size of the building and therefore it will also become capable of housing more animals.