Article wierdness

"It isn't a bowl of lion's eye soup unless it contains the eye of a lion."

This morning it struck me that it's a wierd peculiarity of English to
use the definite article before 'eye' in that context. Lions, after
all, have two eyes, so logic dictates that the article should be
indefinite. However, when a part of a whole is an ingredient in a
recipe, it seems that we use the definite article to refer to the part
(this doesn't apply to measurements - it's "the eye of a lion", but
it's "a cup of water").

Is there a deeper layer of logic here? Has this phenomenon been
studied?

Adrian.