12 days, from Lukla
A variant on the Everest Trek, is the valley west of the Khumbu valley, namely the Gokyo Valley. Like the walk to Everest base camp, the first piece of your hike is from Lukla to Namche Bazaar. Hence the trail runs to the west rising up the Gyoko Velley.
Most hikers make the trip from Gyoko to Gyoko Ri, a 5357m summit. The views from here are magnificent.
You can cross from this valley to the Khumbu valley, the Everest route. For that you need to go over the difficult Cho La pass, 5420 m. This requires a a glacier crossing, making the right equipment imperative.