When travelling into the Everest region, most people either go directly to Everest base camp. The other option, which is less frequented is to travel one valley over to go to a region called Gokyo. You then climb a peak of 5300 (17,700 feet) metres and have a stunning view of Everest which is one valley over.
The best time of day to climb the peak, (which takes about 3 hours return) is to do it beginning at 4:30 am in the morning for sunrise. But alas, you have to get up in the dark and hike in the freezing cold. Or, if you are lucky and the weather is good, you leave in the middle of the day and then watch the sun go down turning Everest a golden colour. That is what we did and we got super lucky.
Here are some amazing photos of Everest and the surrounding mountains. By the time we got down the peak it was very cold but a warm lodge and a cup of hot tea and all was good.