How to obtain the Ktarian Headset
Initially this easter egg was found using a crew which included an immortalized (Lv. 100 FF/FE) Ensign Crusher and Commander Riker. Choose any ship and scan Regula I as many times as needed to receive the Ktarian Headset.
There is no set amount of scans needed to obtain the headset and appears to be a random chance.
UPDATE: The Headset has been confirmed to be available at Risa (located on the Galaxy Map between Federation and Ferengi space). Use an Immortalized Commander Riker, Ensign Crusher and Ensign Robin Lefler assigned to the U.S.S. Enterprise NCC-1701-D (available on all platforms). 
UPDATED UPDATE 3/2021: With the recent changes to scanning and the implementation of Constellations and Polestars, I was able to find the Ktarian Headset at Risa with a different crew and ship than described above, with a regular scan. Then go to inventory and examine the item which will launch game. Tap the frisbees to send them towards the blue vortices. Your level will increase to some level [needs edit]. Tapping the camera icon will change the background turning your device's camera on and off.
How I came to this solution
First I gathered info from people that had been able to get the Ktarian Headset prior to me and it seemed that people had used different ships and so that did not appear to be of influence. Then there was crew: everyone indicated to have Ensign Crusher (***) and Commander Riker (**) active and the consensus seemed that one needed to scan Regula or Regula I.
I started out by having Crusher not at lv 100 and Riker in the vault and scanning Regula I. Nothing happened and I got Crusher up to lv 100 FF/FE and left Riker in the Vault. Again nothing happened for a few weeks and so I got Riker out of the vault on active duty and started scanning Regula I.
I used a Enterprise-D lv 8/9 and also had Lt. Commander Data active (the only other crew I had active that was part of the episode at FF/FE and I do have multiple crew that were part of the episode in the vault). Since other players indicated that they used a different ship and/or did not have Data FF/FE or active I deduced that the ship, its level and or other crew besides Crusher and Riker were of no influence.
With Riker and Crusher active and scanning "Regula I" I was able to get the Ktarian Headset within 2 weeks.
UPDATE: Use an Immortalized Commander Riker, Ensign Crusher and Ensign Robin Lefler assigned to the U.S.S. Enterprise NCC-1701-D and scan at Risa (located on the Galaxy Map between Federation and Ferengi space).