The most common and most likely answer to this crossword clue is the 14 letter word SELF-AWARENESS
I've seen this crossword clue in Independent
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Buddy explains!
The straight part
We know that "Number-one cognitive quality" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case number-one cognitive quality = self-awareness.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.
Crossword Answer definitions
SELF-AWARENESS noun- The state or property of being self-aware; state of being aware of oneself as an individual.
- A personal trait regarding someone's ability to persistently and accurately perceive their presence amongst other people, their own knowledge and abilities; and subsequently, to recognize their limitations for those even which they're particularly confident with.