Cryptic crossword clue
Artist much appreciated struggle by the French to defend refined dish of theirs 11

The most common and most likely answer to this crossword clue is the 11 letter word RATATOUILLE

I've seen this crossword clue in The Guardian
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Buddy explains!

The straight part
We know that "dish of theirs" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case dish of theirs = ratatouille.
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 RATATOUILLE mean?

RATATOUILLE noun
  1. A traditional French Provençal stewed vegetable dish consisting primarily of tomatoes, zucchini and eggplant, with other ingredients.

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