There are four kinds of en passant checkmates.
- The pawn moves, checking the king itself.
- The pawn moves, checking the king, while also allowing another piece to check as well. This is a double check.
- The pawn moves, not checking the king, as to allow another piece to check the king. This is a discovered check.
- The pawn moves, not checking the king itself, as to allow two other pieces to check the king. This is a double discovered check.
Here are the fastest possible games for each category.
Reversing each game, adding on a ply, provides for Black's fastest mates. These are sourced from the same Lichess forum linked in the accepted answer.
[Title "Type #1"]
[FEN ""]
1. e4 f5 2. exf5 Kf7 3. Qg4 h6 4. b3 Qe8 5. Bb2 g5 6. fxg6#
[Title "Type #2"]
[FEN ""]
1. e4 d5 2. d4 dxe4 3. d5 Kd7 4. Bc4 Nc6 5. Bg5 Qe8 6. h3 e5 7. dxe6#
[Title "Type #3"]
[FEN ""]
1. e4 e5 2. g4 Nc6 3. g5 d6 4. Qh5 Kd7 5. Bh3+ f5 6. gxf6#
[Title "Type #4"]
[FEN ""]
1. c3 f5 2. h4 Kf7 3. Qb3+ Kg6 4. Qf7+ Kh6 5. h5 a6 6. d4+ g5 7. hxg6#