Targeting the weakest square (f7) is the quickest.
I have the following solutions, using standard algebraic notation, with white move 1, white move 2; black move
1. Nf3, Ne5; f6 (or f5)
2. e4, Qh5+; g6
3. Nxg6, Ne5#
If 1. ... Nh6 (to protect f7)
2. e4, Bc4; e6
If 2. ... d5, then 3. Bxd5, Bxf7, continuing as below,
while on 2. ... f6 (or f5), 3. Bf7+; Nxf7 4. Qh5, Qxf7#
3. Bxe6, Bxf7+; Nxf7
4. Qf3, Qxf7#
Blocking the diagonal first with 1. ... e6 gives us:
2. e4, Qh5; Nh6
3. Nxf7, Nxh6+ (or Nd6+); Ke7
4. e5, Qf7#
if 2. ... Ke7
3. Qxf7+; Kd6
4. Na3, Nac4+; Kc5
5. Qxe6, Qd5+; Kb4
6. Qa4#
1. ... g6 is no better, with play as in other lines.
2. e4, Bc4; Nh6
2. ... e6 allows 3. Qf3, Qxf7#
3. Qf3, Bxf7+; Nxf7
4. Qxf7+
While 1. ... d6, threatening the Knight, fails to
2. e4, Bb5+; Qd7
3. Qf3, Qxf7+; Kd8
4. Qxf8+; Qe8
5. Qxe8#
1. ... c6, trying to give the king an escape route
2. e4, Qf3; Qa5
3. Qxf7+; Kd8
4. Qxf8+; Kc7
5. Nf7, Qd8#
It appears the best black can do, is get mated at move 7.
1. Nf3, Ne5; Nc6
2. Nxc6, Ne5; e6, continuing as above.
3. e4, Qh5; Ke7
4. Qxf7+; Kd6
5. Na3, Nac4+; Kc5
6. Qxe6, Qd5+; Kb4
7. Qa4#
Or, with 3. ... Qe7, as OP suggests,
4. Nxf7, Nd6+; Kd8
5. d4, Nf7+; Ke8 (if Qxf7, 6. Qxf7, Qxf8#)
6. Bg5, Nd6+; Kd8
7. Qe8#