How does white get the upper hand?

Please tell me if there is anything wrong with the puzzle in any way. It should be thoroughly analyzed enough, though.

Actually, since white already has the upper hand at the beginning of the puzzle, just call it "how white gets in a position to promote pawns and, ultimately, defeat black"

Looks to me like black should play 2...Bxg8 to not overload the king.

Also 1. e6 looks dangerous.

I think e6+ is about the same result, but not much faster in any case. Please tell me if I am wrong!

If you can find one more move for Black to make in this situatuation, another strong move for white would be to advace the knight to C5. It would put presure on blacks knight, and protect the bishop if black chooses not to capture it. If black does decide to capture the bishop, it would put pressure on the rook and knight, and in this scenario, black does not want to be trading pieces. This puzzle was fun, I think you did a great job!

A good move for white can be

as it gives check for king like a deflection