If you played Beast Mastery in Legion, you'll already know Hati - Thorim's wolf that sacrificed himself and became bound to our artifact weapon, Titanstrike. He served faithfully as our secondary pet until his essence was lost when Titanstrike was drained at the end of the expansion. Or so it seemed. In reality he narrowly escaped destruction, but at the cost of his essence being split in two. Patch 8.1.5 introduced a quest chain that allows you to rescue Hati's fragmented essences and reform him. This allows you to tame him in any or all of six different colours. While the quest chain can be started in any hunter spec, Hati is a Spirit Beast, and therefore only usable by Beast Masters.
The story begins at Tradewinds Market, Boralus, for Alliance characters, and The Grand Bazaar, Dazar'Alor for Horde. The Image of Mimiron has a new quest for you - Spark of Genius. As Alliance, The Image of Mimiron is easy to find - he's standing right outside the Snug Harbor Inn. Finding him on Horde is a little more complicated - he's down in the Grand Bazaar (by the docks), on the upper western terrace. Essentially, across the main staircase from the flight point.
This quest leads you to The Storm Peaks where you meet up with Mimiron and your old friend Grif Wildheart in The Inventor's Library, south of Ulduar. During the follow-up quest, Telemetry Online, Mimiron attempts to triangulate the position of Hati's essences, while you fend off enemies that have followed the signal: first waves of vrykul and proto-drakes, and then waves of silithids. These attackers are not particularly hard.
The next quest, The Huntsman's Creed, sends you to the nearby Temple of Storms - the same place where your story with Hati began back in Legion. There you find Thorim incapacitated and part of Hati's essence running amok.
After speaking to Grif you have to fight Hati's essence and reduce it to 21% health, at which point you can absorb it. This event can hurt, and if you don't avoid the lightning you can die. Thorim then offers you the next quest, Paid in Blood, leading you to Silithus.
The other half of Hati's essence is located in Silithus, north of the Ruins of Ahn'Qiraj, which you can reach by taking the portal in Boralus or Dazar'alor and then flying south.
Speak to Grif Wildheart to begin. This fight is a little harder than the first one, as Hati also emits balls of lighting that chase you. If one hits you it will freeze you in place and make it far more likely you'll die from the other lightning spells. After Hati's essence is reduced to 21% health it is incapacitated and you can absorb it.
After absorbing the rest of Hati's essence, your next quest, Spark of Imagination, directs you to rendezvous with Mimiron in the Spark of Imagination in Ulduar. When you arrive don't enter the raid instance, instead take the portal located just outside the entrance which sends you directly to the Spark of Imagination. Note that you'll need an empty stable slot for the next stage. Fortunately, there's a stable master inside the instance, but if your stable master slots are also full then you'll need to abandon an existing pet before you can proceed.
In the Spark of the Imagination speak to Mimiron to start the reformation quest - Bonds of Thunder. If you don't have a free pet slot: there's a YZ-85 Biorearrangement Unit <Stable Master> standing to Mimiron's left, so you can stable an active pet. Once you're ready, place the Thunderspark on the console and sit back and enjoy the show - no interaction is really needed here. Once the cinematic has completed, Hati has returned!. Taming Hati is automatic - at the end of the cinematic you'll have him by your side as a pet.
In the final stretch of the chain, the quest Homeward Bound sends you back to the Temple of Storms to speak to Thorim. This is an important step, as it will allow you to tame other Hati colours. In the quest, Reverberation, Thorim teaches you how to use the Thunderspark (now conveniently placed in the middle of the Temple floor) to summon different known aspects of Hati and them tame them. Hati will not harm you during the taming. It seems that the Thunderspark will remain in the Temple of Storms as a permanent feature, so you can return and tame other colours of Hati later.
Each essence may be tamed as a separate version of Hati. The six possible options are:
The light grey 'Hati's Sacrifice' version is new, and not one of the versions that were previously available with Titanstrike.
Hati retains his name when tamed, but you can change it if you want to.
Not far from Thorim stands Clockwerk a mechagnome who sells consumable toys and accessories for Hati:
Hati as a mount!