What is the correct usage of “of” in phrases about quality [duplicate]

You should say 'How good a cook is she?'.

'Good of a', 'big of a', etc, are informal US regional dialect forms, and not considered standard English.

While noun + of a + noun is very much standard, e.g. an angel of a wife, a nightmare of a day, things get more complicated when adjectives are used. Using adjectives of quantity is standard, e.g. enough of a reason, too much of a coward. Using adjectives of degree — 'good/bad', 'big/small', 'long/short', 'old/young', 'hard/easy', 'near/far', and so on — the 'of a' pattern is not considered standard English.

Merriam-Webster’s Dictionary of English Usage summarizes this dialectal construction as “a fairly recent American idiom that has nearly a fixed form: that or how or too, or sometimes as, followed by an adjective, then of a and a noun.”