B777 pilot reported smoke in the cockpit requiring oxygen masks to be donned. Flight crew performed inflight engine shutdown and diverted to suitable airport.

Narrative

Had gone to rest in the bunk for about 1 hour when I woke from a shuttering feeling. The Relief Pilot #2 opened the crew rest door and we saw smoke. Enough smoke that it was getting hard to see. We immediately went to the cockpit and donned oxygen mask. The crew was in the process of a drift down around FL380 and going through to checklists. XXXX was squawked and ATC was addressing with [priority handling]". One checklist concluded with the left engine being shut down. There were comms with Dispatch and ATC. We then navigated to ZZZ1 while getting weather and running checklists. At first ZZZ1 wasn't reporting a ceiling or wind. So ZZZ2 was considered. Dispatch was asked if there were any other airports to consider. About 10 min later ATC was able to report the weather at ZZZ1; which was closer and had a longer runway. Everyone discussed and decided it was the best choice. Continued to ZZZZZ and smoke in cockpit dissipated and held for the GPS XX. Held so we could lose the remaining altitude and once we had the altitude commenced the approach. Landed; stopped on the runway and had the fire/rescue look over the damaged engine. They cleared us for taxi via a follow me. Continued to park where we had to wait for power as the APU was MEL. Cause: Unsure at this point. The left engine had been replaced recently."