Solution:

Congrats to @samostically to find the first move and some of the key variations!

So how can white win?

The correct move is 1. Qe6!! which threatens to give checkmate on d7 or f7.

Obviously black should not take the queen with one of its rooks, because then 2. c8Q# respectively 2. g8Q# finishes the game.

Furthermore, white wins with 2. Rd8# respectively 2. Rf8# if black takes the queen with 1. ... dxe6 or 1. ... fxe6.

Let's focus on the two main variations now:

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a)

1. ... Nf6 which prevents the checkmate on f7 and protects the pawn on d7.

2. Rxf6 Nd6
3. Rxg6 dxe6
4. g8Q+ Kd7
5. Qd8#

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b)

1. ... Nd6 which prevents the checkmate on d7 and protects the pawn on f7.

2. Qxg6! Here 2. Rxd6? loses because of 2. ... Qe2+! 3. Kxe2 exf3+. 2. ... Nxg7
3. Qxg7 e5
4. Qh8+ Ke7
5. Qd8+ Ke6
6. Qf6#