I left the aircraft recommendation blank on purpose for this one, because that's part of what I found so hard about making this trip. I crashed two times before perfecting the route you see here. Your plane, whatever its make and model, will need to accomplish each of the following in turn:

1. Take off from a tiny gravel air strip at high elevation (hint: back up to the grass to give yourself extra room).
2. Quickly climb to at least 20,000 feet.
3. Be able to cruise at around 25,000 feet.
4. Deftly navigate within the narrow canyon of Lukla (elevation 9,400 feet) to figure out how to land there.

A couple of obvious hints about aircraft selection: I doubt a jet will work (I saw one guy try and he hilariously failed at Lukla). A Cessna 172 only has a 14,000 foot ceiling so obviously that won't work either. It's gotta be something in between. 

The views are obviously spectacular, from A to Z. The route is hard enough in clear weather, so I suggest choosing clear skies, but take on harder weather if you wish. I would not bet on your survival though. :)