The noun filet first emerged in the early fourteenth century. For a while, there were a bunch of accepted spellings, including felet, fyllet, filett, and fillit, but around the time when the word was verbified at the start of the seventeenth century the most common forms were narrowed down to fillet and filet. It comes from the Middle French word filet, which had the rather interesting definition of "ribbon". The connection there is that early styles of the dish were prepared by being tied with a string. Further back, we can trace it to Latin filum, which meant "thread", and that traces to a Proto-Indo-European word sounding like gwhi and with the same meaning. This makes fillet related tot he words file and profile, but those are stories for another time.
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