I think the key is
To promote your pawn to a Queen first, and control the A8-H1 diagonal with the Q-B combo, as @PerpetualJ has mentioned.
The sequence of moves would probably start with
1. a6! (push the pawn) Bf4 (mobilize the Black bishop). We then need to block the black square diagonal g1-a7, since this is the best diagonal for Black to block the coronation with 2...Be3 next. Then we use a pawn to achieve this, using 2. b6. 2. ... cxb6 is then forced, otherwise 3. b7 crowns the pawn. Then 3. Kb5!! blocks the Black bishop from accessing a7. Black rushes the pawn with 3...e4 4. a7 e3. White needs to control the diagonal as we mentioned, they do this with 5. Bh1. Black keeps rushing with 5...e2 6. a8=Q e1=Q. Unfortunately it’s now too late for Black, because the Queen serves as a barrier for the Black King now! 7. Qf3+ forces 7...Kg1, and then the final blow is delivered with 8. Qg2#.
So I think the answer is
Mate in 8 (lol, it rhymes!)
The first Black move 1...Bf4 with the intended 2...Be3 is necessary; moving the two pawns out of the way instead loses because 1. a6 c5? 2. a7 e4? 3. Bh1 creates an unstoppable a-pawn. 1...Bg1 has some merit, but ultimately fails because the Black king has to move out of the way; Ke3, Ke2, and Ke1 block the e-pawn; and Kf1 has the same effect as above except blocks the king in faster with 7. Qg2#.