There are four Business Class award seats available every day in June 2015 on Japan Airlines' flight from Los Angeles to Osaka, Japan. Hat tip to reader Oren for passing along the tip.

The award space is bookable with American Airlines and US Airways miles much for as low as 50,000 miles each way. There are excellent connection options to and from Los Angeles and Osaka, so this space effectively connects all of the United States with all of Asia.

Unfortunately the seat servicing the route is angled lie flat.

There is excellent award space in Japan Airlines First Class between San Francisco and Tokyo for most of the next 11 months.

There are a lot of cool things you can do with miles. One of the coolest things I've thought up, and one which I am strongly contemplating booking, would turn 57.5k United miles and 10k Avios into:

A trip to a Pacific atoll to be determined, Guam, and Japan. Three places I've never been.
Fourteen hours on the famous Island Hopper route.
Flat bed business class for seven hours.

The center piece for this plan is the Island Hopper route, which I became obsessed with last week and read everything a google search could find.