You can acquire the Pheasant item by hosting a hunt.
To Host a hunt, you must build a Hunters lodge as well as a Hunters yard and set of Mews to store and generate hunting animals.
These buildings are all available from level 5 and can be built by your manor.

Once you have acquired some falcons, tap on the hunters lodge and press the button beneath "falconry display" . This will initially reward you an amount of honour but come the end of the hunt, you will be rewarded with a pheasant.

Different hunts provide different rewards. Later items include rabbit, boar and stag. Some of which are also used for cooking and others for crafting decorations.