Tromposky with 2...g6

"Best" is all relative.  What may be the best move at one level. may not be the best at another.

Supposedly so...afterall , you played that to double up his pawns.  Then after he takes, you move e3.  But it might not hurt him much, once he moves that front doubled pawn forward, opening up his Bishop.  (But.  Stockfish shows c3 the best move)  

I play the black side. I like to fianchetto my KB. After BxNf6, I'm having difficulty to equalize or take the initiative.

I assume the bishop pair would compensate for the doubled pawns.

1.d4 Nf6 2.Bg5 d5! Black should be fine. No need to fianchetto the bishop here, which is usually done by people who feel morally obliged to play the King's Indian Defense against everything.

This is what I play against the Trompowsky as Black, and so far no one's lopped off on f6 after that. They transpose to the Richter-Veresov. If you want to fianchetto, you can always follow up with Nbd7 and g6, Bg7. Works very well.

Thanks BonTheCat. I'm indeed a fanatic of KID and Gruenfeld Defense. And would love to obtain the bishop pair. 2... g6 follow White's plan.

White will willingly concede the bishop pair in most Trompovski lines.

2...g6 3.Bxf6 exf6 4.e3 with the plan g3, Bg2, Ne2, 0-0 and c4 is quite annoying to meet: Black has no realistic attacking chances, and white enjoys a space advantage and a very solid position.

I have played 2...d6 a couple of times OTB, which more or less results in the same structures: White has a safe, small positional advantage.

I tend to play it Vere-Slav fashion (...c6 and follow up with ...b5 and a queenside attack), rather than Grünfeld style. However, given that White has placed a knight on c3, hitting White's centre with ...c5 should work, too.

Generally speaking, as Black against the Trompowsky 'proper' (i.e. anything but 2...d5 and 2...Ne4), I've always felt that Black wants to avoid exf6 if at all possible. Black's pawn formation is left a bit static, and it's hard to get any real attacking chances. The reason I started playing 2...d5 is that Black on Bxf6 can set up Rubinstein's (typically as White though) favourite pawn formation f7-f5-d5-c6. Presumably this is also why most White players avoid it and prefer to transpose to the Richter-Veresov, QGD or some other QP opening.

Thanks IM for your lesson. So... 2... Ne4 is the best reply?

I (if you pardon me for butting in!) would say that the best reply against the Trompowsky is according to your own tastes - what you feel comfortable with. 2...Ne4 can become extremely sharp, and you really need to know what you're doing there. One reply which is generally considered to be one of Black's best options is 2...c5, but it leads to a game which may not be to everyone's liking (1.d4 Nf6 2.Bg5 c5 3.Bxf6 gxf6 4.d5). I played it for many years with fair results - I typically got a decent game out of the opening and the positions which arise from it are fascinating. I gave it up basically because I felt it was time for me to rejig my opening repertoire as Black, and I gradually moved away from 1...Nf6 in favour of 1...d5 in reply to 1.d4. This led me to take up 2...d5 in reply to 2.Bg5 whenever I played 1...Nf6.