You need one Auchenai Phantasm and 6 healing cards for the OTK. So you need to draw seven cards in total, which means the earliest OTK is turn 3 on the coin and turn 4 if you go first.

Mana-wise, you have 4 mana to spend in both cases, which is sufficient for Phantasm plus two Flash Heal; the rest of the damage has to come from four (or five) Desperate Prayer. Therefore when you mulligan, keep every Prayer in your opening hand and either toss all the rest or keep at most one of each of the other cards.