I'm seeing a lot of fares out of Europe to North America on Icelandair A/E class (Premium Saga) for under $1200. I'm sure there are many city combinations available, but the best I have found so far is
TXL to SEA for around $1180 (1084 EUR). Flights book into A / E class.
Dates and destinations seem to be wide open. You can also book directly with Icelandair for this price and include a free stopover of up to 7 days in Iceland.
Note that Icelandair's business product features recliner seats similar to domestic first class or international premium economy. They are not lie-flat. However the flight times coincide well with day-time (I won't be sleeping) and the connections at least for my
TXL BER -> SEA itinerary were short at 1:30.
Slightly more expensive deals to the west coast have been available lately on all major carriers from select cities (namely SOF and RIX) with very long layovers. This the cheapest flight I have seen with good availability across Europe and reasonable layovers.