How can white make progress here?

H5 to H6 for starters

That loses a pawn to Nxg6. If white follows with hxg7, black recaptures with the queen and simultaneously moves the lady into dangerous attacking position along the open seventh rank. White is almost in zugzwang. The only move that doesn't lose on the spot is Ka5. Nh2 loses because it allows black to play Qe5.

... but black can't do anything either. A very noble journey by the white king (Kb6-a5-a4-a3-a2-b1-c1-d1-e1-f1-g1-h2-h3-h4-g5-f5-e6-e7-f8-g8) will eventually win the exchange Nxg6 hxg6 while black will be forced to alternate between Kc7 and Kc8 to avoid losing pieces. Now the white king will return to whence he came (via Kg8-f7-e6-f5 etc.) and force the zugzwang win for white.

Excellent puzzle, Nordlandia.

King to a5 then to a4, a3, a2, b1, c1, d1, e1, f1, g1, h2, h3, h4, g5, f5, e6, e7, f8, g8 which forces 19...Nxg6 (otherwise 20.Kxh8  21.h6 and 22.g7 wins) then 20.hxg6 wherein you backtrack the White King all the way around to b6 again which puts Black in zugzwang and his position collapses.

I actually posted the above before I saw the solution.

<NJCat> That's very well done! I wouldn't have thought of that

But now that you mention it, it would seem that you only considered the first HALF of the manoeuvre. 

After going to g8 and forcing the knight to sacrifice itself with panic on g6, it is now time to return to the king's homeland, on a5, but NOW, having somehow gained a tempo, we will get to b6 when it is BLACK's turn, who will then be forced to spoil his position, as the king will no longer have the c7 square. 

The king can gain a tempo on his way back by going to BOTH e7 and f7 (the latter square was not available before, as it was guarded by a knight on h8). 

<AussieRookie> me too :-) we must have all (including NJCat) seen the light at one and the same moment!

Yes, I had to edit my original comment once I truly figured it out.

On the way back, the White King can get from g8 to e6 in 2, 3 or 4 moves, so you can always gain a tempo.

Cool puzzle, thanks!

1. h6 is actually a blunder equivalent to a rook. 

Engines are not able to see the zugzwang theme, - (modified) yes they detect the king can eat the knight but not returning to force zugzwang. 

Ka5

Ka5