B757 Captain reports high oil temperature due to low oil quantity at FL400. As the temperature climbed into the red; the engine was shut down and the flight returned to their departure airport.

Narrative

Just after leveling off FL400 I noticed the left engine oil temperature in the amber band; oil quantity zero. I asked the First Officer to get the checklist out for high engine oil temperature and I would fly the airplane. The engine oil temperature high checklist requires landing at nearest suitable airport. ZZZ was the nearest suitable airport due to a snowstorm over the mountains and my unfamiliarity with the mountain airports. We declared an emergency; got lower altitude and clearance to ZZZ. I asked the First Officer to call Dispatch. Then we noticed that the oil temperature was climbing and was now in the red band which required shutdown as directed from checklist. I asked the First Officer to get the checklist for shutdown and had Dispatch standby until engine shutdown was complete. We completed the engine shutdown checklist; then I talked with Dispatch. I advised him of our situation; that we were returning to ZZZ as it was the nearest suitable airport. I advised the Purser that we were returning to ZZZ with touchdown in approximately 20 minutes and that our engine was shutdown. I told him that we had requested equipment to be standing by; but I expected a normal landing and taxi-in. I also made PA and told passengers we were returning and upon landing they would see emergency equipment standing by when we land but it was a precaution; expect a normal landing. I landed the airplane and taxied in with no further problems.