What the achievement asks
Take Soridormi up on her offer to pet the fluffy fur you picked up in a Time Rift.
Blizzard ties this one to the character rather than the account, so it does not carry over to an alt the way most of the branch does.
It is a joke achievement, and it is also one of the two in this branch that a player can chase for days without ever getting a chance at it.
The gate
The fur comes from Ulderoth Mutagen Orbs, and those only appear during a rift with Titan invaders. In any other reality there are no orbs, so there is no buff and nothing to do.
Open your map in Thaldraszus while a rift is running and hover the rift icon. It names the invaders for that hour. If it does not mention Titans, this one is not available and you can spend the twenty minutes on Just Following Chronological Orders instead.
The three steps, once the right hour arrives
Find an orb. Ulderoth Mutagen Orbs are medium-sized yellow spheres sitting on the ground during the first phase of the rift. They are easy to see once you know to look. Players report them most often around the large Titan keepers that wander the edge of the Reservoir, so fly the perimeter rather than staying in the middle of the crowd.
Open orbs until you are offered Fluffier Fur. Each orb offers you a choice of buffs, in the style of Torghast or the Cobalt Assembly. The one you want is Fluffier Fur, which cheerfully tells you your cuddliness is increased by 100%. It is not guaranteed on the first orb, or the third. In a well-populated rift there are plenty of orbs and you can open five in a few minutes.
Go back to Soridormi with the buff still on you. She is at 51.5 57.1, where the rift starts. She notices the fur, asks whether she might, and you say yes.
Do not lose the buff on the way back. It ends when the rift does, and a few players have reported reaching her only to find it gone. Once you have Fluffier Fur, walk straight to her and finish it before doing anything else.
If the orbs are not spawning
A rift with few players in it produces few orbs, and several people have reported flying around an empty Titan rift for a whole event without seeing one. The fix is the same as for the rest of this branch: a busier rift, at a busier hour. In a full group the orbs are numerous enough that finding Fluffier Fur is a formality.
Where it fits
One of ten under Closing Time, which rewards the Bestowed Sandskimmer and is a piece of A World Awoken.