Hi
How do you win this
If white plays as inaccurately as two of the respondents have analyzed. A third has the accurate method or White to win.
Sack of Potatoes post (#28) has Black throwing away the draw on move 8 (Ke7 draws), White throwing away the win on move 10 (Kf6 wins) and Black throwing away the draw on move 11 (Ke8 draws).
If you depend on the defender misplaying the position then even poor moves can work out for the attacker.

This is how I would solve it.
You would be wrong then.
1.e3?? Is already a draw.
Your 4th move for Black is wrong! 4...Ke5 draws!
There are 2 and only 2 correct answers.
1.Kd2 or 1.Kf2 works. Nothing else does.
White must get his King to the 4th rank with the pawn still on e2. If it is on e3, Black draws with opposition.
With the pawn left alone on e2, once you reach the position of BKe6, WKe4, Wpe2 with White to move, you have the move e3!, gaining the opposition necessary to win the game!

WHite wins by-
1.Kd2-Kf7. 2. Ke3-Ke6. 3. Ke4-Kd6. 4.Kf5-Kd5. 5.e4-Kd6 6.Kf6-Kd7 7.e5-Ke8. 8.Ke6-Kf8 9.Kd7 Kf7 10. e6-Kf8 11.e7-Kf7 12.E8=Q+ (+-)
By Torsha Kundu, 2011, India.

If black to move, it´s a draw. The BlackKing just stays in front of the pawn and prevent the white King to come in front of the pawn with opposition. If it´s white move than be more than careful with checks of the black king. You have to move more with the King (shure, King in front of the pawn). In the example of NaranjoManuel2010 move 5 e5+ is the big mistake. Now the white King is never going in front of the pawn (6. ... Ke6 and it´s a draw) (Rule for Black: Stay on the same line as the white pawn or stay in opposition to the white King). In DefenderPug2 it´s move 4. e4+. Why should black go to g5? He will go to e5 and again it´s a draw. White will never manage to come in front of his pawn. Always be careful with pawn moves, they never come back

Jetoba wrote "When Black is played inaccurately it is easy to find a win for White." I would say, the very opposite is true: If white playes accurately it is definitely a win for white, no matte what black is doing.

How to win…
You need to move your king first. You can note that the king’s opposition is pointed for black, but the e-pawn will make the difference.

And something post 36 left out is why the pawn must remain on e2.
After 1.Kd2 Kd8 (instead of 2...Kd7 - Now BLACK has the opposition) 2.Kd3 Kd7 3.Kd4 Kd6 4.Ke4 Ke6 and then here is the kicker ... 5.e3!!. The opposition now belongs to White. 5...Kf6 6.Kd5 or 5...Kd6 6.Kf5. In both cases, White wins!
Sack of Potatoes post (#28) has Black throwing away the draw on move 8 (Ke7 draws), White throwing away the win on move 10 (Kf6 wins) and Black throwing away the draw on move 11 (Ke8 draws).
If you depend on the defender misplaying the position then even poor moves can work out for the attacker.
In @Sack_o_Potatoes's variation the black king started on e7 instead of e8, so the game is drawn from the start.
Sack of Potatoes post (#28) has Black throwing away the draw on move 8 (Ke7 draws), White throwing away the win on move 10 (Kf6 wins) and Black throwing away the draw on move 11 (Ke8 draws).
If you depend on the defender misplaying the position then even poor moves can work out for the attacker.
In @Sack_o_Potatoes's variation the black king started on e7 instead of e8, so the game is drawn from the start.
Yes, the first error I listed was Black throwing away the draw, not White throwing away the win. Yet the analysis made it look like a White win. Either bad analysis or trolling.
This is how I would solve it.
Interesting.
It does allow Black to force a draw with different moves for 3, 4 and 8. It also allows Black to maintain the draw it could already force with the moves it played on 5, 6 and 7.
When Black is played inaccurately it is easy to find a win for White.