"Eats" is a noun.
The word glutton is a noun. A glutton is one who eats excessively.
Like most words, "eating" can be used as different parts of speech. It is a noun when used like this: Eating is fun. It is a verb when used like this: She is eating a hot dog for lunch.
The part of speech for this particular word is a noun.
part of speech
The part of speech for "answer" is a noun.
what part of speech is beneath
adverb
The part of speech for "explicit" is an adjective.
The part of speech for "lecturer" is a noun.
The part of speech for "rapid" is an adjective.
The part of speech of "came" is a verb.
The part of speech for "breezy" is an adjective.