It depends on the placements of the kings, but generally you have to keep corresponding squares and zugzwang in mind.
How do I best handle this situation with pawns? (diagram)
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Well, and then? 8. Ke7 Kg7, draw. To win, you have to do an end run around the other guy's pawn blockade, possible only with favourable king placements.
I think the OP was asking how to handle the Stonewall pawn structure.
I mean very favourable, like
With that pawn structure, whoever moves a pawn first loses, and you can't force Zugzwang because his king can defend the keystone pawn from two squares so he always has a move. The only way to win is to get there first.
Assuming OP is referring to these pawns and not a specific position:
Nothing too general can be said since it very much depend on the position. The e6-pawn is weak (cannot be protected by a pawn), so attacking it can be good and there is an outpost for white at e5 which is an excellent square for the knight to manouver to. The same thing can be said that black should do. If the position is closed you are typically manouvering your pieces to good squares and look for pawn breaks.
Ive noticed this kind of thing pop up in a few games and it just seems to shut down huge sections of the board. What do I do when this happens?