B737 Captain reported the trailing edge flaps got stuck at 15 degrees during flap extension on final approach. The flight crew performed a go-around; used an alternative solution that deviated from the QRH as it was not applicable; and safely landed.

Narrative

During Captain OE (Operating Experience); we set up for an autoland into ZZZZ as the weather was 1;200 ft. RVR. After a thorough brief and setup; we decided to land flaps 40 for a little more visibility. During flaps extension on final approach; the pilot monitoring selected flaps to 30°; then 40°; and ran the Landing Checklist. At that time the pilot monitoring saw the trailing edge flaps were stuck at 15° and verbalized the configuration. The pilot flying spun the speed bug back up so we didn't get too slow and said during the debrief he noticed the PFD's (Primary Flight Display) speed tape looked different that normal; but hadn't realized why until the pilot monitoring called out the failure. A missed approach was executed and [priority handling was requested]. Once the missed was completed and the aircraft stable; the pilot flying continued flying and took radio duties; while the pilot monitoring ran the QRH for Trailing Edge Flaps Disagree. That concluded that the flap handle did nothing to alter the flap position.Because of the weather in ZZZZ; the crew discussed going to the listed alternate airport of ZZZ. However; with the current fuel on board and the flaps stuck extended; decided that would not be prudent as we would land in a very low fuel state. While conducting the QRH; the procedure set us up for a flaps 15 landing. With the weather still at 1;200 ft. RVR in ZZZZ and an autoland required; we elected to use the alternate flap extension system to attempt to get the flaps to 30 as we didn't know how the aircraft would react to a flaps 15 autoland; or even if it would do an autoland at all. We deviated from the QRH to make this happen as it seems the procedure never planned on a requirement of a CAT III autoland after a flap malfunction. The flight attendants were briefed with the brief items; and a passenger announcement was made. Once realigned with final; the flap handle was moved to the 30 detent; the alternate extension was used; and the flaps and slats extended. The pilot monitoring changed the landing speeds to flaps 30 speeds and an uneventful autoland was accomplished on Runway XXL. ZZZZ ARFF (Airport Rescue and Firefighting) was standing by to do an inspection which concluded with no issues and the aircraft taxied to the gate under its own power. The Chief Pilot and Dispatch were called and debriefed.