Well, while @DrXorile found the answer, I'd figured out the key idea and the first couple of moves of the actual answer, but then my attempt is quite different. In the end, I understood why it doesn't quite work, but it still involves a really beautiful sequence and requires sharp play by Black. So here it is.
First thing, none of Black's pieces can move except the e1 rook, meaning that capturing this rook without check will result in stalemate.
This gives us Black's strategy to annoy us: keep checking our king with this unprotected rook, as we cannot capture it, until the 50-move rule applies. Essentially, we cannot allow Black to deliver check.
On the other hand, this black rook is not completely free: leaving the 1st rank would result in Ra1#. For instance, if White starts with a promotion, Black cannot capture the promoted piece immediately because of this threat.
Now to the point: the idea is 1. e8=N!! threatening a mate from a completely different angle: 2. Nd6 and 3. Nb7. Black's rook cannot deal with both mate threats.
The only answer by Black
to avoid 2. Nd6 and having to deal with two mate in 1 threats is 1. ... Rg1 threatening check (and therefore a draw). White answers with 2. Kh5, preventing any checks. The threats remain for Black: again, capturing the rook would make Nd6 Nb7 unavoidable, and leaving the first rank without check is instant death. Keep threatening check remains the only option.
Black answers with 2. ... Rc1 or any rook move not leaving the 1st rank and threatening check.
3. Nd6 Now both mate in 1 threats are in place, but check is coming. However, White is ready to take it as they have a secret plan. 3. ... Rc5+
4. Kh4 and for the next few moves Black has only one way to keep checking the white king, and absolutely needs to do it.
4. ... Rh5+ 5. Kg3 Rh3+ 6. Kf2 Rf3+ 7. Ke2
Now Black has essentially two options, and both lead to its fall for different reasons.
$\bullet$ Either black only checks the white king from the top or from the right, and in this case Black cannot stop White from reaching the b1 square, e.g.
7. ... Re3+ 8. Kd2 Re2+ 9. Kc1 Re1+ 10. Kc2 Re2+ 11. Kb1
7. ... Re3+ 8. Kd2 Rd3+ 9. Kc2 Rc3+ 10. Kb1
$\bullet$ Or black checks white from below at some point, e.g.
7. ... Re3+ 8. Kd2 Re2+ 9. Kc1 Re1+ 10. Kc2 Rc1+
$\bullet$ If Black checks white from below, the idea is
11. Rxc1! You can capture the rook without stalemate as you set the pawn free by doing so, and the pawn promotion doesn't come with check.
11. ... h1=Q (not done yet as the queen covers both mating squares, b7 and a1!)
12. Nb7+ Qxb7 13. Ra1#
12. Ra1+ Qxb7 13. Nb7#.
$\bullet$On the other hand, if White reaches b1, then you have the magnificent sequence 11. ... Re1+ 12. Ka2 Ra1+ 13. Rxa1 h1=Q 14. Nb7+ Qxb7 15. Kb1#.
Now the reason why this sadly fails is that Black can ensure having their rook on the d file when Kb1 is played:
7. ... Re3+ 8. Kd2 Rd3+ 9. Kc1 Rd1+ 10. Kc2 Rd2+ 11. Kb1
and then instead of 11. ... Rd1+, Black will simply capture the knight as the other threat of Ra1# is temporarily blocked by the white king.