See New in Chess Yearbook #90, Larry Kaufman.  This is dated but has some interesting analysis on these lines.  It leads to dynamic interesting positions.  

See: https://www.chesspub.com/cgi-bin/chess/YaBB.pl?num=1236255050/3

I used to play it a lot but mainly play McCutcheon these days.

1. e4 e6 2. d4 d5 3. Nc3 Nf6 4. Bg5 de4 5. Ne4 Be7 6. Bf6 gf6!?

This has been mostly off my radar. When I looked at the French with 3Nc3, I looked primarily at the Winawer, and also the MacCutcheon.

My old notes show that 6...gxf6  has received theoretical attention for a long time. For example Byron Jacobs devoted a chapter to it in his 2001 book French Classical (unavailable in digital form):  "Recapturing with the pawn on f6 leads to an altogether more dynamic and double-edged game than the solid 6...Bxf6." Pedersen also looked at 6...gxf6. Does anyone know if this was covered by Aagaard and Ntirlis?

It is also part of Ivan Cheparinov's 2023 repertoire for Black available at Modern-Chess. It is his reply to 1. e4 e6 2. d4 d5 3. Nc3 Nf6 4. Bg5.

Other good sources include Chess Publishing and the Opening Encyclopaedia from Chess Base.

According to Chess Assistant, Black has had very good results overall from this position. However for 2024, the results were just average. To know more would require some digging.

6.Nxf6 is much less common, but my notes show that it was recommended by IM Mihajlov at Chess24. Black almost always recaptures with the bishop. However, 6...gxf6 has been played in a few games. Answering both 6Bxf6 and 6Nxf6 with 6...gxf6 seems consistent, but I have some doubt about 6Nxf6 gxf6 for Black.

Actually, scratch that. I found that chessbase has a DVDrom by Knut Neven called French with Nc3. I think I'll get this one. It's not one of those with video commentary, which is good because I prefer a "computer-book" format, having had very good experiences with Mihail Marin's Catalan and English and Rogozenko's Dragon DVDroms. But I hope this one on the French is good - I heard they vary in quality 

As is the case with more and more variations, there really is no better source on the gxf6-Burn than Chesspublishing.com, in this case McDonald's French site which has always been excellent.  

You could play through Neil's annotated games to get a feel for the positional and tactical themes and which lines are currently played on GM level, and then consult with a database and Pedersen and/or Psakhis' books to fill in whatever few cracks remain to form a complete Black repertoire.

NB: A White repertoire with 4.Bg5 is suggested in "Chess Openings for White, Explained". The focus there is on the lines 7.Nf3 a6 8.Qe2!?, 7...f5 8.Nc3 Bf6 (here the currently popular 8...a6! goes unmentioned, which makes it all the more attractive as a Black repertoire choice) 9.Qd2 c5 10.d5, and 7...b6 8.Bc4 intending Qe2 and 0-0-0. Black needs to be well-prepared against these quite critical lines.

I really want to learn the Burn ...gxf6 varition. It's so dynamic! Unfortunately this comes at a price in that if Black doesn't play with accuracy it can easily backfire   So can anyone recommend me a book which delves deeply into variation? Thanks

Edit: I ahve the Pedersen series btw.