B-777 Captain reported after takeoff an engine showed an EGT exceedance. The pilots returned to the departure station.

Narrative

I was Captain of Aircraft X; Day 0. We departed ZZZ at XA:25 on a rescheduled delayed departure flight to ZZZZ. There were four pilots on the augmented flight. During engine start we got a Configuration Gear; TO configuration flaps EICAS during engine start. After about :10 seconds the EICAS went away. The relief pilot said he had seen a EPR spike during start and we concluded that the EPR spike had been a glitch and set off the EICAS to think we had applied takeoff power. Systems returned to normal and we continued.The departure was normal and uneventful. I was pilot flying and we used Runway XX with a slight tailing crosswind. During climb on the ZZZZZ2 departure we were turned to the north. As I had flaps retracted and accelerated I noticed a EICAS THRUST ASSYMETRY TAC. I pointed to the EICAS and continued climbing as I directed the FO (First Officer) to complete after takeoff checklist and run the checklist for the TAC (Thrust Asymmetry Compensation). At that point we noticed a red box around EGT on right engine. The engine was operating normally at that time but was about 70 degrees higher than the left engine. The red box showed we had a EGT limit during the climb. We continued climing to a safe altitude and requested a level off at 14000 and a vector clear of traffic as we worked a checklist. While ATC vectored us around traffic we got another EICAS Eng EEC (Electronic Engine Control) Mode R.The checklists had us reset the EEC switches to ALT. The relief pilot mentioned that the Eng Lin/Surge/Stall u checklist would be appropriate as we had a indication expedience. We reviewed that checklist and concluded the engine was stabilized.At this point after crew discussions and completing the tasks on the checklists I made the decision that we should return to ZZZ and land. We had stayed within 30-35 miles from ZZZ as we worked the situation. We contacted dispatch on Satcom and patched to Maintenancel. A discussion of all the faults and situation concluded in agreement with ZZZ landing as safest plan. I discussed with Maintenance that I would rather not dump fuel and if we could safely land overweight we would do that. We had Dispatch send the performance data to conclude that a safe 30 degree landing could be made on Runway XXC at ZZZ and also ensure a single engine go around with performance if it was needed. Our weight allowed this and we briefed Runway XXC; for ILS approach. We [requested priority handling] with ATC and requested clearance to Runway XXC.After a briefing we were vectored to join Runway XXC. I slowed the aircraft early and configured to ensure limits and speeds were safe. A heavy landing has some differences and handling is challenging. I used Autobrake 2 and Flap 30 to decrease stress on the aircraft. The landing was normal landing and I gently applied brakes and we were able to exit normally only about 2/3 of the runway. Fire crews inspected the plane and we continued to the gate.