Cryptic crossword clue
Doctor’s right first off, assuming Royal Society fellows may wear these

The most common and most likely answer to this crossword clue is the 12 letter word MORTARBOARDS

I've seen this crossword clue in Independent
M
O
R
T
A
R
B
O
A
R
D
S

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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.

What does MORTARBOARDS mean?

MORTARBOARDS noun
  1. A square board, with a handle, on which mortar or plaster is carried: a hawk.
  2. An academic cap that has a flat square top with a tassel.

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