Scale and fillet the mullet, and remove the line of small bones that runs down the middle of the fillets using a pair of pliers.
Remove the seeds from the peppers and cut them into 3 mm/⅛ inch dice. Peel the carrot and dice the same size as the peppers. Dice the courgettes to a similar size but do not peel.
Heat half the oil in a frying pan, then add about 25 g/1 oz of the butter and allow it to sizzle but not brown (do not allow the pan to get so hot that it burns the butter). Season the fillets of mullet on the meat side only, and gently place them in the pan, skin side down, for about 20 seconds, then turn and slowly cook on the other side for about 2 minutes or until cooked. Remove them from the pan and drain on kitchen paper. Cover and keep warm.
Tip the fat out of the pan, add the remaining oil and gently fry all the vegetables for about 10 seconds. Add the vinegar and the fish stock, return to the boil, then remove the pan from the heat and gradually add the remaining butter, shaking the pan until all the butter has melted.