Not a proper problem

The situation below is simple enough: White to play and win. White has nine possible moves, several of which win. The rest do not.Your mission, should you choose to accept it, is to find the winning plan.

White to play: Find the winning plan

8/pp1k2b1/8/3PPK2/6P1/PP6/8/8 w - - 0 1

This isn't a proper problem, but it is a good exercise. I reached this position playing out a position from my game against Jim from last Saturday. My opponent, as usual these days, was Stockfish.

Now for a subtle, confounding hint: It appears that if Black's bishop were on h8 instead of g7, then the game is probably a draw. I'm still trying to work that out, as well as the whys and wherefores. I plan to post the answer next week sometime. Hopefully I'll understand the position a bit better then, else wise it'll be more of a variation dump. Also, I hope I'll be done with this game by then!