For example, "жареный цыпленок" is animate.
Furthermore: Words such as глаз ("eye"), рот ("mouth"), нос ("nose"), таз ("pelvis; hipbones") and still others refer to body parts of an animate person, and are made of living tissue, and are even masculine in gender. So, logically, one might expect that such words have an accusative identical in form to the genitive -- but they don't. Yet, as it-ogo notes, a (dead) and roasted young chicken IS animate! (I'm reminded that Karl Orff's Carmina Burana has an entire song performed by a Жареный Лебедёнок, "roasted cygnet"!)

P.S. I couldn't think of an example of a masculine body part that one might use in referring to a roasted chick or swan -- maybe окорок, "thigh"? If so, then one would say Я съел жареного лебедёнка (animate masc. acc.) but Я съел жареный лебединый окорок (inanimate masc. acc.)