I believe the answer is:
One piece (the king), in six moves, which reiterates Glorfindel's answer.
This was tested using the Stockfish.js 10 Engine on Chess.com. Most of these were tested up to 40+ depth.
Here are a few setups I tested (mate in x moves denoted by "+Mx"):
> One piece: (best: mate in 6)
Ke8: +M6 1.e4 Kf7 2.Qg4 Kf6 3.Qd7 Kg6 4.d4 Kf6 5.Bc4 Kg6 6.Qf7#
> Two pieces: (best: mate in 7)
Ke8, a7: +M7 1.e4 Kd7 2.Qg4+ Kd6 3.Qg7 Kc6 4.Nc3 a6 5.Nd5 Kd6 6.Qc7+ Ke6 7.Qe7#
Ke8, b7: +M7 1.e4 b5 2.Bxb5+ Kd8 3.Qf3 Kc7 4.Qf6 Kb7 5.Qd8 Ka7 6.Qc7+ Ka8 7.Bc6#
Ke8, c7: +M7 1.e4 Kf8 2.Qg4 Ke7 3.d4 c6 4.Qc8 Kf6 5.Bc4 Kg6 6.Qg4+ Kh7 7.Qg8#
Ke8, d7: +M7 1.e4 d5 2.Qg4 Ke7 3.Qg7+ Kd6 4.exd5 Kxd5 5.Qg6 Kc5 6.Nc3 Kd4 7.Qd6#
Ke8, e7: +M7 1.e4 Kd7 2.Qg4+ Kd6 3.Nc3 e6 4.Qg7 Kc6 5.Nb5 Kb6 6.Qc7+ Ka6 7.Qa7#
Ke8, f7: +M7 1.e4 Kf8 2.Qh5 Ke7 3.d4 Ke8 4.Bc4 Kd7 5.Qxf7+ Kc6 6.Bf4 Kb6 7.Qc7#
Ke8, g7: +M7 1.e4 Ke7 2.Qg4 g5 3.Qxg5+ Kf7 4.e5 Kf8 5.Bb5 Kf7 6.Qh6 Ke7 7.Qf6#
Ke8, h7: +M7 1.e4 h6 2.Qh5+ Ke7 3.Qxh6 Kd7 4.Bb5+ Kc7 5.Qc6+ Kb8 6.Ba6 Ka7 7.Qb7#
Ke8, Bf8: +M8 1.e3 Ke7 2.Qg4 Bh6 3.Bb5 Kf7 4.b3 Bg7 5.Qf5+ Bf6 6.Bb2 Kg8 7.Qxf6 Kh7 8.Qg7#
Ke8, Nb8: +M9 1.e3 Ke7 2.Nc3 Nc6 3.Nd5+ Kd6 4.Qf3 Ne5 5.Qf6+ Kxd5 6.e4+ Kxe4 7.Qe6 Kd4 8.c3+ Kc5 9.b4#
Ke8, Ng8: +M9 1.e3 Nf6 2.Bb5+ Kf8 3.Qf3 Kf7 4.b3 Ke7 5.Bb2 Nd7 6.Qf5 Kd6 7.Qxd7+ Kc5 8.Qc6+ Kb4 9.Bc3#
Ke8, Bc8: +M10 1.d3 Be6 2.Qd2 Kd7 3.Qh6 Bd5 4.Nc3 Bc6 5.Nf3 Kc7 6.Nd4 Ba8 7.Ndb5+ Kc8 8.Qg7 Kd8 9.Bg5+ Kc8 10.Qc7#
> Three pieces: (best: mate in 7)
Ke8, a7, h7: +M7 1.e4 Kf8 2.Qh5 h6 3.Qe5 Kf7 4.Bc4+ Kg6 5.Qf5+ Kg7 6.Qf7+ Kh8 7.Qg8#
Ke8, a7, e7: +M8 1.Nf3 Kd7 2.Ne5+ Kd6 3.d4 a5 4.e3 Kc7 5.Qf3 Kb6 6.Qc6+ Ka7 7.Ba6 a4 8.Qb7#
Ke8, d7, e7: +M8 1.e3 Kf7 2.Qf3+ Ke6 3.Qf4 Kd5 4.Nc3+ Kc6 5.Qb8 d6 6.Bb5+ Kc5 7.Qc7+ Kb4 8.a3#
Ke8, Bg8, h7: +M9 1.e3 g6 2.Qg4 Ke7 3.Qxg6 Kd7 4.Bb5+ Ke7 5.Nc3 Kd8 6.Nd5 Kc8 7.Qe8+ Kb7 8.Bc6+ Ka7 9.Qa8#
> Four pieces: (best: mate in 8)
Ke8, a7, b7, h7: +M8 1.e4 Kd7 2.Nc3 a6 3.Qf3 Kc7 4.Nd5+ Kb8 5.Bxa6 bxa6 6.Qf7 Ka8 7.Qc7 h6 8.Nb6#
Ke8, a7, g7, h7: +M8 1.b3 g5 2.e3 Kf7 3.Qf3+ Ke6 4.Bc4+ Ke7 5.Ba3+ Kd8 6.Ba6 Kc7 7.Qb7+ Kd8 8.Qe7#
Ke8, Bf8, e7, g7: +M9 1.e4 e5 2.Qh5+ g6 3.Qxg6+ Kd8 4.Bb5 Ke7 5.Nf3 Kd8 6.Qe8+ Kc7 7.Qc6+ Kb8 8.Ba6 Ba3 9.Qb7#
> Five pieces: (best: mate in 8)
Ke8, a7, b7, g7, h7: +M8 1.e4 a6 2.Qh5+ Kf8 3.Bc4 Ke7 4.Qf7+ Kd6 5.Qxb7 Kc5 6.d4+ Kxc4 7.Qc6+ Kxd4 8.Nf3#
Ke8, Bf8, e7, f7, g7: +M10 1.e3 e5 2.Bb5+ Ke7 3.Qh5 g5 4.Nf3 e4 5.Qxg5+ f6 6.Qf5 Kd6 7.Qxf6+ Kc7 8.Qc6+ Kb8 9.Ba6 exf3 10.Qb7#