Traditional 雞 had a bird held by the sound element 奚; simplified 鸡 keeps a hand 又 gripping a bird 鸟 — the barnyard fowl you can actually catch. The rooster has crowed in Chinese script since the oracle bones.
The Rooster is a zodiac year, and 鸡 feeds the language: 鸡蛋 egg, 鸡肉 chicken meat, 母鸡 hen. 落汤鸡 'chicken dropped in soup' is anyone caught in the rain. Kung Pao chicken 宫保鸡丁 may be the most-ordered 鸡 on earth.
A hand 又 holding a bird 鸟 — the CHICKEN you caught for dinner.
China's map is said to look like a rooster — schoolchildren learn the northeast as the head and Hainan as a foot. The 鸡 is literally national geography.