In Orvis's book Favorite Flies one can read that it's belived that a fly called Bright Brown was the original of the Fiery Brown we know today. Bright Brown was a fly described by Charles Cotton in the book The Compleat Angler, published 1653. Side note: there is a digitalized copy of the book which can be read here.
Charles Cotton wirtes the fly is best used in March, and should be dressed thussly: the dubbing for which is to be had out of the skinner's lime-pits, and the hair of an abortive calf, of which the lime will turn to be so bright as to shine like gold ; for the wings of this fly the feather of the brown hen is best". Later he writes of December, saying: "Few men angle with the fly this month, no more than they do in January: but yet, if the weather be warm, - as I have known it sometimes in my life to be, even in this cold country, where it is least expected, - then a brown that looks red in the hand, and yellowish betwixt your eye and the sun, will both raise and kill in clear water." This sentence, "red in the hand , and yellowish betwixt your eye and the sun", has become rather famous, and an enigma. It is used to describe the body of the true Fiery Brown fly; "but it is one of those things no man can find out." There are a few interesting notes to be made about Charles Cotton's description of how the "old Fiery Brown" fly should be dressed. An attentive reader probably stopped for moment when he describes how dubbing should be taken from of an abortive calf. This was a fairly common practice at that time, and one can read in other historic texts following: "Refort to a Skinners Lime-pit, and there get the Hair of various colours, and you may get most excellent dubbing of skins of Calves and Colts, that are Tewed ; and several colours and shades of one skin [...] that have been neatly Tewed in Skinners Lime-pit".