B737 Captain reported a flap asymmetry during takeoff resulted in return to departure airport and safe landing.

Narrative

[Requested priority handling] (flight control malfunction); while in Mountainous terrain and thunderstorms within the vicinity of the airport resulting in an air-return; Long Tarmac delay; parked at a non-operational gate; difficulty communication and coordination; with ZZZZ [airport] operations: difficulty communication with company (using other peoples; cell phones):While taxiing to runway XX; we completed the Before Takeoff Checklist"; including the Takeoff configuration". As we received position and hold on runway XX; we obtained an amended takeoff clearance to turn on a heading of 140 and climb and maintain 4;000'. Take off roll was normal; and immediately after liftoff; we received a momentary "takeoff warning". I promptly verified that the speed brake handle was fully forward in the retract position as I remembered this was an issue with the micro switch; and I also verified that the flap handle was in the Flaps 5 detent. The takeoff warning only lasted approximately 3-4 seconds; and we continued to climb to the assigned altitude and heading. At this time; I noticed that the flap indicator had a split; I believe the right flap was at 5; and the left flap was less than 5 with a definitive split. I instructed First Officer to turn on the autopilot. I immediately informed the tower that we had a mechanical issue with the flaps and would return to the airport; so I [requested priority handling]. We requested vectors to give us time to work on the checklists; coordinate with the flight attendants briefing and send a message to dispatch via the "Divert" function on the ACARS. First Officer continued to fly the aircraft and communicate with the tower while being vectored. I ensured that one of us remained in "Terrain" due to the terrain in the area while the other had "WX radar" active due to the storms near the airport. Fortunately; the initial heading kept us away from any terrain. I proceeded to the Non-Normal checklist and determined the issue was "(NG) Flaps Trailing Edge Flaps Asymmetry" Non-normal Checklist. I petitioned ATC that we needed the straight in ILS Z XX. They asked if we were ready to proceed; and I replied; "No;" as we needed vectors (time) to prepare. While First Officer continued as pilot flying and I pilot monitoring; I completed the following necessary tasks: Non-normal checklist; loaded the approach and extended off the farthest waypoint; set VREF 40+30 plus wind additive; re-cruised the VNAV; briefed the approach; briefed the FAs; etc. Upon completion; I stated that I was ready to transfer controls to execute the landing. As I took control; I instructed the First Officer to double-check and verify that I completed the necessary checklists; including the landing distance assessment due to the high approach speeds and runway conditions. We also asked for braking action reports; which were reported as "Good" by an Airbus 320. I asked for the "DEFERRED items" checklist and gave the Double Chime indicating imminent landing. We informed tower ATC that we were ready to commence the approach and to have equipment standing by. We broke out about 600' on the ILS and landed smooth and firm with Max reverse thrust and MAX Autobrakes. We exited the runway at taxiway 1 and requested the equipment to follow us to the gate to ensure no issues with our brakes or tires. I also requested a thermal scan of the brakes; to which they replied that it was "normal". Operations had us park at the gate; which was not operational. After completing the parking checklist; I made the necessary ELB (Electronic Log Book) right-ups; one for the Trailing Edge Flaps Asymmetry Log and another for the overweight landing log. Interestingly; the flap indicators at that time showed both flaps with no split at flaps 2; our detent that was dictated in step 1 of the Non-normal checklist. I had a debrief with the First Officer and thanked him for his expertise. I contacted Operations; who informed me that they were bringing portable stairs to the aircraft. I explained to them we were at a non-operational gate. Therefore; I requested a gate change with a working jet bridge to offer egress. After the stairs were installed and the main cabin door opened; the gate agent and mechanic wanted to speak to me and entered the flight deck. The mechanic wanted the failure details; and the gate agent requested "on/in" times. After talking to the gate agent and the mechanic; I went in the forward entry and made a PA to the passengers; briefly explaining what had happened without giving too many details. I kept it simple to reassure them and mentioned that we were in the process of getting a new gate. I asked the flight attendants to give the passengers whatever snacks; drinks; or water. I used the First Officer's phone to call Dispatch. In hindsight; I learned that I should have called the Chief Pilot first. The Chief Pilot called the First Officer's phone to speak to me. (The first officer has an international data plan; while I did not). I do not recall the name of the Chief Pilot I spoke to; I should have written it down. After talking to Dispatch; I talked to Maintenance Control via the mechanic's phone and gave them the details of what happened. Throughout the time on the ground; I had difficulty communicating with the ZZZZ [airport] Operations due to language barriers and their slow response as I insisted on a gate. Maintenance continued to complete the tasks set out by Maintenance Control. I asked if the mechanics physically looked at the flap transmitters; wiring; rigging; etc.; and their reply was they were doing a BITE (Built In Test Equipment) test via the avionics bay while running through each of the flap settings. As time progressed (it was approximately an hour since I blocked in); I kept requesting a gate due to the long tarmac delay. Another gate finally opened; but we had to wait for another air carrier to vacate. When it was time; I told the passengers to take their seats; and we were towed to the gate. Then; all the passengers deplaned. Maintenance continued running tests and eventually signed off the paperwork. As my adrenaline began to drop; I realized that I was physically and emotionally drained from the experience. I asked the First Officer how he was doing since I did not think it was best for me to continue. He conveyed that he was fit to continue. A deadheading pilot; a Captain B; offered to inquire to see if he was legal to fly. I decided to call Crew Scheduling to inform them of the situation and request a pilot change because I was physically and emotionally drained. Scheduling asked when I would be fit for duty; calling out numbers; i.e.; 10hrs; 11hrs; etc.; frankly; I was so drained; I replied 10. They scheduled me to deadhead back to ZZZ [airport] the next day. I asked if I could fly back to ZZZ later that day. Then; I realized I could ride the jump seat on the flight; basically swapping positions with Captain B. Scheduling and I conferred and chose the jump seat option on the flight. They instructed me to notify them when I landed in ZZZ [airport]; which I did later that evening. I asked Capt. B if I could fly in the jump seat and he said yes. The gate agent offered me the Deadheading captain's seat as an option; but I elected to remain in the jump seat just in case any problems developed again; I felt I could assist. Once we got confirmation that Capt. B was legal and could fly the trip; I went up to the gate; received my jump seat pass; and made a public announcement to the passengers that I would no longer be the captain of the aircraft and Capt. B would be in charge. We flew back to ZZZ without any further issues. I reached out to the Chief Pilot after I arrived in ZZZ."