The Outbreak Perfected exotic quest was secretly added a while ago, but if you're developing a new character you'll definitely want to take the time to complete the quest to unlock this superb weapon. To kick off the secret scavenger hunt leading to the Outbreak Perfected exotic weapon, head to Titan first.
Begin the Titan Heroic Adventure, Bad Neighbors
Once you encounter the first room with the shrieker (those floating hive enemies that spit out a barrage of purple lasers) rather than head straight through into the next room, turn left and open the door. If you can't open the door, there are some prerequisites players are still untangling. It also might just be a bug.
Once you're in the hidden room, examine the Fallen device on the desk to pick up the Fallen Transponder. It's pretty difficult to miss.
Finish up the quest and head back to orbit.
Find the nodes
Next you'll need to hunt down nodes that correspond to the areas hinted at in each branch of the Fallen Transponder, which are located on the EDZ and Nessus. This video by Niitq usefully details each of the six node locations.
Return to the farm
Don't worry, I forgot it exists too. Once you've found every node, fly to the farm and head to the basement of the main building. As soon as you spawn in, turn right, head into the basement and talk to the Fallen captain hiding down there. This will launch the time-limited Zero Hour mission.
Complete the Zero Hour mission
This is a race against the clock. You have 20 minutes to complete some combat rooms, a lengthy maze and jumping puzzle section, and a final boss room stuffed with enemies. It's a 690 power mission and enemies are protected by a lot of shields.
Once you've spawned in and slipped through the wall via a vent to your right, you're thrown into the first combat zone featuring some Fallen captains and lots of tracer shanks (there are a lot of these in Zero Hour). If you have it, Arbelast is effective at range and strips shields efficiently. Supplement your loadout with a good sniper rifle like DARCI, or a decent roll on the Black Armory sniper rifle, Tartara Gaze. You have to kill everyone here to bring down the blue shield and proceed to the old Tower courtyard.
The second combat area is populated by more tracer shanks and a few massive shanks. Snipe what you can from the ruin of the wrecked corridor that you take on the way into the courtyard, then focus down the big shanks. Ahead and on the right edge you will find a path that leads around into a corridor, which takes you into the Tower interior.
Inside the tower you will find clusters of Fallen protected by Servitors, overlooked by long-range Vandal snipers and yet more tracer shanks. Counter-snipe the snipers, nuke super the clustered servitors and Fallen, and then lay continuous damage on the Walker at the end of the room. Once that's dead, the next door will drop. Fight through two small rooms full of bog standard Fallen, and then proceed to the maze.
It's easier to watch the jumping puzzle section to see the route through this convoluted area. Planet Destiny does a clear run through at this point in his guide video.
After this you get to meet one of Destiny 2's most famous villains. TR3-VR is an aggressive cleaning bot that patrols the basement, chopping up anything it finds. This is terrifying, but the solution is simple. Turn left and hug the left wall, take any left turns that you find, and hit any switches you can find. Once you've hit four the exit will open and you can progress to the final combat room.
Here it's just a great big fight. Wipe the enemies, kill the boss and claim your Outbreak Perfected.