Hearthpwn

Not a Beast in sight! (Until the Raptors swarm from the deck, that is.) Basically the idea is to use the 1-Drop Elemental generation cards to passively complete the quest and forgo the usual Beast synergy. In other words, this deck wants the Quest reward without sacrificing board tempo. You could even toss aside the Quest in your opening hand if you wanted to, just make sure you keep enough 1-Drops in hand at the ready for when the Quest is drawn again.

Edit 1.1: Decided to add in some Thunder Lizards to take better advantage of Elemental synergy earlier in the game, meaning the deck now does run two Beast cards. And by changing Hunter's Mark to Unleash the Hounds + the eventual Beasts earned for fulfilling the Quest, I decided to make some room for Kill Command.