eggshake vs milk shake

eggshake and milk shake both are nouns.

Word	Noun	Adjective	Verb	Adverb
eggshake	Yes	No	No	No
milk shake	Yes	No	No	No

As nouns, milk shake is a hypernym of eggshake; that is, milk shake is a word with a broader meaning than eggshake:

eggshake: a milkshake with egg in it
milk shake: frothy drink of milk and flavoring and sometimes fruit or ice cream

Other hypernyms of eggshake include milkshake, shake.

eggshake (noun)	milk shake (noun)
a milkshake with egg in it	frothy drink of milk and flavoring and sometimes fruit or ice cream