Honing Instincts

I found two fairly effective ways to do this quest, depending on which zone is the Black Empire Invasion. In both cases, wait to summon your hatchling until you're at the location of your choosing and ready to start.

If Uldum is currently under attack, either go to the Lost City of Tol'vir (/way 62.02 70.70) or to the Akhenet Fields (/way 58.48 51.65) where the Foul Observers spawn and kill as much as you can. For Tol'vir, if the summoning event is up you can keep resetting that (kill everything, extinguish the candles, go kill a few other random mobs until it pops up again). If you choose to kill the Observers, you also have a chance of spawning the rare Gaze of N'zoth which can drop [item=175142], needed to make the toy [item=175140].

If the Vale of Eternal Blossoms is under attack, go to the Big Blossom Mines (entrance at /way 70.07 58.42) and start killing all the mantid you see. By the time you've made your way through the mines and into the vault, then cleared both rooms down there, things should be respawning back at the start. I managed to clear the whole place and loop back to the start with a couple of minutes to spare, netting me a few more kills on the respawns before my little hatching withdrew.

Both of these methods netted me around 25-30% of the experience bar, obviously results will vary depending on how fast you're able to kill things. Lastly, I recommend doing this quest with Warmode off as if you die you will lose your shadowbarb friend.

Killing anything while your Shadowdrone is following you increases the progress bar to the next stage. This is not a quest you want to just insta complete.
I spent the full 10 minutes killing every mob I could find in Tol'vir raised the bar about 30%.
Helpful Tip: If you are flying with the pet following you and something hits you, the pet will aggro to the mob and will despawn when you fly too far away.

Edit: The progress bar can only go to 4999/5000 while doing this daily quest. Upon completion, it will transition to the next stage, but still requires a daily reset to pick up the next stage quests.

Does not persist through logout, so if you get disconnected for any reason, you get screwed out of the rest of your buff timer.