I've heard of mullet being caught on fly; most, I suspect, are thinlips. Bob Cox caught a thicklip from Essex (Crouch or Blackwater, I think) on a sandeel imitation streamer-fly; but subsequent efforts were unsuccessful despite there being plenty of fish in the area. I've tried half-heartedly in the Medway, but without so much as a follow. Mike Ladle catches thicklips on the Dorset coast on maggot flies (polyethylene body baited with live maggots) when mullet are feeding on seaweed maggots. But that's "using fly tackle" rather than "fly fishing".
Techniques such as fly fishing are obviously worth trying, and are certainly of academic interest, but for consistent success with thicklips I think we are best advised to stick with bait fishing. Even bait fishing is not without problems, though. Mullet demand good presentation, which means relatively fine tackle, but they fight tremendously hard and tend to live in snaggy environments (rocks, weed, pier piles etc), which makes the use of fine tackle hazardous. Answers on a postcard, please...!