Clue: Artist much appreciated struggle by the French to defend refined dish of theirs (11)
Answer: RATATOUILLE
Explanation: Definition: dish of theirs
 RA(abbrev. for for an artist who is a member of the Royal Academy of Arts) + TA!(short for "Thank you!"/"Much appreciated!") + TOIL(a struggle/a hardship) **plus**(by) LE(French for the article "the") **containing**(to defend) U(letter denoting things refined/upper class).
Defn: …/of the French, perhaps prepared by a rodent chef of the same name: