Does the structure "an example of" take a plural or a singular noun or both?

Problem: cheese is a mass noun, so is kind of really neither singular nor plural. Mass nouns take singular verbs, so we'll pretend like it's singular!

Next problem: "an example" is singular, and so we'd expect the thing that is the exemplar to also be singular. Ria is an example of a girl. check! Ria are examples of a girl. wrong!

Sentence one and two are grammatical, but with slightly different nuances. 1. indicates a sort of "ex plurality": "protein rich food" itself means the sum of all foodstuffs that are rich in protein. 2. indicates a sort of "ex singularity": "a protein rich food" is more specific than "protein rich food". It's one out of many. If your only point is to say that cheese has protein, then either choice is valid. If your point is to say that cheese has protein comparable to all other protein rich foods, then pick 1. If your point is to say that cheese has protein comparable to specific protein rich foods, such as dairy, then pick 2.

Sentence 3. is ungrammatical because of the number mismatch. In this case, you've got is in its use as copula along with a predicate, cheese, and the predicative expression. The number, singular or plural, ought to be the same on both sides of the equation.

You could say Cheeses are examples of protein rich foods. Here, cheeses is a plural count noun meaning "different kinds of cheese".