The text plays on the double meaning of the words 'kinky' and 'sick'. 'Kinky' typically refers to unusual sexual preferences, and 'sick' can mean both 'disgusting' and 'ill'. The humor arises from the unexpected escalation from using a feather (implied to be a part of a kinky act) to using the whole chicken, which is absurd and considered 'sick' in both senses of the word.
Kinky : Using A Feather . Sick : Using The Whole Chicken