B737-300 First Officer reported erratic EGT indications climbing through FL310 resulted in stick shaker and rapid bank angle increase. The flight crew recovered from the stall and returned to landing at departure airport.

Narrative

The Captain and myself arrived at the aircraft at XA55L. The relieved crew reported to us that the First Officer (FO) side VOR/ILS was MEL'd. They said that they had to conduct a missed approach earlier in the day at ZZZ due to a loss of all ILS indicators on both sides. The flight mechanic had run a test in ZZZ that the Captain side system passed. We prepared the aircraft for our first leg to ZZZZ1 and departed around XB55L. The Captain was Pilot Flying (PF) and First Officer (FO) PM. During the climb as we passed through FL310 we observed a steady decrease in the #1 engine EGT. This decrease continued down to zero before the display blanked out. A few seconds later; the EGT gauge began to fluctuate erratically; ranging between zero and 1250 degrees Celsius. All other engine indications remained normal. We leveled off at FL310 and executed the memory items for the engine limit; surge; or stall checklist. Even with the #1 thrust lever closed; the EGT indications continued to act erratically.We began to work through the QRH checklist and elected to halt prior to shutting the affected engine down. This was due to the problem most likely being an indication issue and our location over the ocean. At this point; the aircraft was in LNAV/VNAV without the autothrottle engaged. While attempting to troubleshoot the indication issue; the aircraft started to execute a left turn as per the active route in the FMS. This turn resulted in a rapid increase in bank to about 25 degrees; and a rapid loss in airspeed as the autopilot attempted to maintain altitude. We received a stick shaker activation. The PF disengaged the autopilot; and applied forward pressure and right aileron inputs. The aircraft was in a stable descent by FL300. The PF had the PM set a descent speed of 280 kts.We asked for an immediate return to ZZZZ; and [requested priority handling] with ZZZZ Center. We attempted to contact dispatch using XXX.X but could not get a response. As we got closer to ZZZZ; the PM set the aircraft up for the ILS Runway XX. We planned to do a flaps 15 landing and utilized the performance in flight section of the QRH to calculate landing distance due to the abnormal flaps condition and the overweight condition. As we were receiving vectors to the ILS; we overshot our initial intercept of the localizer to the left side of the Runway XX centerline. Initially; the localizer was alive on the Captain's side through the overshoot. We received a vector back to the localizer. At this time; we were in IMC. Once we received a re-intercept heading; we attempted to intercept and track the approach. At this point we lost ILS course and glide slope indications on the Captain's side. The PF attempted to follow the flight director and inadvertently descended a couple hundred feet below our ATC cleared altitude of 2500 ft. We were still a couple miles outside of the final approach fix. We broke out of IMC shortly after. I'm not sure of the altitude that we broke out at but it was between 2000 and 2300 ft. AGL. The aircraft was now located about 1 NM to the right of the Runway XX centerline. There was some confusion as to where exactly the airport was due to the erroneous indications of the flight director. The flight director continued to show a turn to the right. ATC queried about our position on the localizer and we requested a visual approach to Runway XX. This was approved and we made a safe landing in ZZZZ around XC10.