Who hasn't bonded over a chicken or two in their day?
Lady Chatterley and Mellors bond over the chickens that Mellors has set nesting on some pheasant eggs in a clearing in the woods. And Connie identifies with these chickens so strongly that they become a symbol of her own captivity. Like her, they have a "soft nestling ponderosity" and a "female urge" (10.37); like her, they lose themselves in fulfilling their female role: "There's no self in a sitting hen" (12.111). When Connie first sees the chickens, her heart breaks because "she herself was so forlorn and unused, not a female at all" (10.36).
Of course what she really wants is Mellors's hands all over her. He handles the pheasant chicks with "sure gentle fingers" (10.52) that calm the chickens down. These chickens, and Mellors's control over them, seem to symbolize the potential animal/female nature that the modern world has suppressed in Connie. They're a symbol of what she could be—lost in nurture rather than lost in her head.