If the boy wants to eat cheese on just one occasion, which I think is the implication, should it not be να φαει ?

Hm, where can I report a mistake? The word "to" (να) is not a preposition here; it is part of the infinitive form of the verb. A preposition must usually be followed by a noun (the object of the preposition) or an implied noun, not a verb. "Να" plus the verb is usually the way that the infinitive is formed in Greek, just as "to" followed by the verb is the way that the infinitive is formed in English. I am by no means an expert in Greek, so would like confirmation of this. Thanks.

In English, a single word can be used in a variety of ways. For example, the word "inside" can be used as a preposition (She colors inside the lines.) or as an adverb (I'm going inside now.) The word cannot be classified as a preposition or as an adverb until it is used in a sentence. The word "to" is another word that can be used as a preposition (I'm going to the store) or as something else - most often as part of the infinitive form of a verb (I'm going to run tomorrow.) . The word "to" is not a preposition in this sentence; it is part of the infinitive. My understanding is that in Greek, the word "να" is not a preposition (it does not come at the beginning of a prepositional phrase!); it is part of the infinitive, just as "to" is part of the infinitive in English.

So does this mean that on Greek, infinitive verbs are also appropriately conjugated??

is this one of those cases where the noun doesn't take an article or is it just me? τυρί doesn't seem to have any article at all coming before itself, is there a reason for that?