Brewer's: Cheese

Tusser says that a cheese, to be perfect, should not be like (1) Gehazi, i.e. dead white, like a leper; (2) not like Lot's wife, all salt; (3) not like Argus, full of eyes; (4) not like Tom Piper, “hoven and puffed,” like the cheeks of a piper; (5) not like Crispin, leathery; (6) not like Lazarus, poor; (7) not like Esau, hairy; (8) not like Mary Magdalene, full of whey or maudlin; (9) not like the Gentiles, full of maggots or gentils; and (10) not like a bishop, made of burnt milk. (Five Hundred Points of Good Husbandry.)

A cheese which has no resemblance to these ten defects is “quite the cheese.”

Bread and cheese.

Food generally, but of a frugal nature. “Come and take your bread and cheese with me this evening.”

A green cheese:

An unripe cheese. The moon made of green cheese. A slight resemblance, but not in the least likely. “You will persuade him to believe that the moon is made of green cheese.” (See above.)

`Tis an old rat that won't eat cheese.

It must be a wondrously toothless man that is inaccessible to flattery; he must be very old indeed who can abandon his favourite indulgence; only a very cunning rat knows that cheese is a mere bait.

Cheese

Something choice (Anglo-Saxon, ceos-an, to choose; German, kiesen; French, choisir). Chaucer says, “To cheese whether she wold him marry or no.”

Now thou might cheese How thou couetist [covetest] to calme, now thou  Knowist all mi names.

P. Ploughman's Vision.

It is not the cheese.

Not the right thing; not what I should choose.

He is quite the cheese

or just the cheese- i.e. quite the thing. By a double refinement we get the slang varieties, That's prime Stilton, or double Gloster —i.e. slap bang up.

Source: Dictionary of Phrase and Fable, E. Cobham Brewer, 1894

Cheeseparer

Cheek by Jowl