3rd update Two more at
13 1/2
[FEN "4k3/pppppppp/8/8/8/8/PPPPPPPP/4K3 w KQkq - 0 1"]
1.f3 f5 2.g4 Kf7 3.Kf2 Kg6 4.g5 h5 5.Kg3 f4+ 6.Kh4 d5 7.e4 a5 8.e5 Kf5 9.
d4 g6 10.h3 e6 11.c4 c6 12.b4 b5 13.c5 a4 14.a3 *
[FEN "4k3/pppppppp/8/8/8/8/PPPPPPPP/4K3 w KQkq - 0 1"]
1.g4 h5 2.h3 f5 3.e4 Kf7 4.Ke2 Kg6 5.Ke3 h4 6.g5 Kh5 7.Kf4 e6 8.e5 d5 9.f3
g6 10.d3 d4 11.c4 c5 12.b4 a5 13.b5 b6 14.a4 *
Updated update Note that this has some similarities to @Retudin's solution even though it was found independently.
Shaved off another move to make it.
13 1/2
[FEN "4k3/pppppppp/8/8/8/8/PPPPPPPP/4K3 w KQkq - 0 1"]
1.h4 f6 2.g3 h6 3.f4 Kf7 4.Kf2 Kg6 5.Kf3 Kh5 6.Ke4 g5 7.d4 g4 8.Kf5 d5 9.
e4 b5 10.e5 c5 11.e6 a5 12.b4 c4 13.c3 a4 14.a3 *
Update simpler one in
14 1/2 moves:
[FEN "4k3/pppppppp/8/8/8/8/PPPPPPPP/4K3 w KQkq - 0 1"]
1.d4 e5 2.Kd2 Ke7 3.g4 b5 4.g5 b4 5.g6 b3 6.a4 h5 7.Kc3 Kf6 8.Kb4 Kg5 9.
Ka3 Kh6 10.h4 a5 11.c3 f6 12.f4 c5 13.f5 c4 14.e4 d6 15.d5 *
And the original one in
17 1/2 moves:
[FEN "4k3/pppppppp/8/8/8/8/PPPPPPPP/4K3 w KQkq - 0 1"]
1.c4 Kf8 2.c5 Kg8 3.c6 Kh8 4.cxb7 h5 5.f3 c5 6.Kf2 c4 7.Kg3 Kh7 8.Kh4 c3
9.b8=B cxb2 10.g4 b1=B 11.g5 Bf5 12.Bf4 f6 13.e4 e5 14.exf5 exf4 15.g6+
Kh6 16.h3 d5 17.d4 a5 18.a4 *
Replay