Stockfish took a long time with this one, but with some help it eventually got to here.
Looks like it's approximately checkmate in 15 from your position. By move 9 (following Stockfish recommendations) it finds a checkmate in 6. Here's what Stockfish says is the best line:
[FEN "4k3/pppppppp/8/8/8/8/PPPPPPPP/RNBQKBNR w KQ - 0 1"]
1. c3 b5 2. Qb3 c6 3. Qa3 a5 4. Qxa5 e5 5. Qc7 f6 6. Qc8+ Ke7 7. Qg8 e4 8. Qxg7+ Ke6 9. g3 f5 10. d4 e3 11. Bh3 b4 12. Bxf5+ Kxf5 13. Qf7+ Kg5 14. Bxe3+ Kg4 15. f3#
Here's a sample game, me vs Stockfish. Took me 18 moves: https://lichess.org/AJFXfWmCQB5I
And another, also 18 moves: https://lichess.org/Ibno7F6cdfhT
I missed some sequences both times.