It's a mistake to view the pawns themselves as overextended targets that you might get to "pick off" as you say in your post. After all, Black's main objective with the pawn storm is probably to open up lines to attack your king, and so your snapping up such overextended pawns really only furthers Black's goal.
As already pointed out in comments in this thread, the strong engine Stockfish recommends the move
1.f4 in your diagram position. And when I opened your post and looked at the position, that was my almost immediate reaction as well. Why? Not because I'm even remotely as strong as Stockfish, of course; rather, I was just reflexively applying a classic positional maxim:
An attack on the wing is best countered by play in the center.
That advice really goes double in this position, where the opponent's king is still in the center, and would need a few moves to get out of there. (Moreover, since Black has overextended on both wings, her king ultimately doesn't have anywhere all that safe to go.) In particular,
1.f4 carries a direct threat of capturing on e5, and if the knight on f6 gets forced to move, White is ready to capture on f7 with dire consequences for Black in the form of a vulnerable king open to attack in the middle of the board. Black's best response is probably nipping any ideas of
fxe5 in the bud with the en passant capture
You might object that we've given Black what I claim she wanted out of her pawn storm, providing a half-open g-file to attack our king, but the point is that we can show Black's pawn storm to be premature by continuing to follow the above maxim. After e.g.
2.Nxf3 Bd6, we just continue to pry at the center with
3.c3! Note that Black is nowhere near getting an attack off the ground, but we've again given her something concrete that she needs to deal with immediately:
4.cxd4 is a serious threat because of an
e5 pawn fork in the air.
Bottom line: Black's pawn storm is an overextension, but not because those pawns are now ripe for the taking. The real flaw in Black's play is that she's been pushing pawns for an attack that she's not yet ready for. That's why the proper response is to fight back in the center, demonstrating that she should have been spending her time developing some pieces first and taking care of her own king's safety. Here are just a few short variations, some featuring very poor Black moves just to illustrate the threats; but even in the lines where Black plays more sensibly, note that in the final positions, White is fully developed and ready to play in the center (and against Black's king), while Black is in no position to harass White at all.
[fen "r1bqkb1r/2p2p2/2n2n1p/1p2p3/p2pP1p1/P2P2B1/BPPNNPPP/R2Q1RK1 w - - 0 1"]
( 1...Bb7?? 2.fxe5 Nh7 3.Bxf7+ )
( 1...exf4 2.Nxf4 )
( 1...Bd6 2.fxe5 Bxe5 3.Qe1 )
2.Nxf3 Bd6 3.c3! dxc3
( 3...Bg4? 4.cxd4 Nxd4 5.Nexd4 exd4 6.e5 )