Which Pawn capture?

The game started as below:

In the game, I played fxg6, planning to claim the f-file with 0-0.  OTOH, hxg6 doesn't leave e6 weak, and retains the option of playing f6 later to open the center (and follows the general guideline of capturing toward the center.

Thoughts?

In general it is better to capture toward  the center.

Which pawn structure would you rather have?

In an end game, the first is obviously better.  However, in the middle game with locked center pawns, the second appears much better for King  safety if I castle short.  Rook on the open f-file is a bonus.  After hxg6 however, I'd likely be tempted to forego castling, since a R on the open h-file seems necessary to guard the flank, and useful if I get an opportunity to rush the g-pawn against White's King.

I'm for sure no French Defence expert

But I do know the LSB might benefit from the f7 square. g6 pawn might need to move (and this is just a possible option not the main plan). Maybe reasoning to support taking with f-pawn.

I'd think f5 would make a better place for the knight. Might look into that.

+1 Agreed. As noted by the next diagrams too, ...hxg6 results in Black having two "pawn islands" while the other capture results in three "pawn islands." For purposes of pawn structure, you typically want fewer pawn islands and so ...hxg6 is generally preferred.

Both recaptures are more or less equivalent. In terms of the pawn structure and endgame prospects 8...hxg6 is better, but 8...fxg6 is better in terms of the open f-file and king safety after 10...O-O.
Instead of 8 Bxg6, trading white's good bishop for a knight, 8 h4 was more powerful. For that reason 5...Ng6 was not a good idea and 5...Nc6 was better.

I agree that Bxg6 was not White's best move...that B is one of White's best pieces against the French.  Nbc6 was the Engine move here, but Nf5 is also evaluated better than Ng6.  I was a bit over-eager to unblock my Bishop, so I moved the Knight prematurely.

I often don't castle at all as Black in the Advance French because the K-side can be so tough to defend...opening the h-file just seems far to risky for me.  However in the above position, if I can build a nice shelter in the middle, hxg6 allows me to challenge White's center with f6.  Both moves have definite benefits.

The engine called fxg6 "best", but gives hxg6 a higher evaluation by a whopping 0.04.

- I would probably take with the f- pawn, although I think both recaptures should win for Black. Actually the game is objectively level, but someone who has just played 8.Bxg6?? must be a total patzer.

Blitz game between players rated below 1400.  Not a huge surprise that there was some sub-standard play.

TotaLLY AGREE. TAKE THE H PAWN.

I'd take with the f pawn, which is often the better recapture when White has a pawn at e5 despite the general "capture toward the center" rule. The reason is the g5 square. After ...hxg6, White will have an outpost square at g5, because driving a piece away with ...f7-f6 loosens Black's K side pawns too much.

After ...fxg6, Black doesn't have that problem because he has ...h7-h6 to control g5. The weakness at e6 is hard to exploit because White's LSB is gone and he can't maintain a N on g5. Also, castling is quite safe for Black now, after which the R will be on the open f file.