Getting from the Sonoran Desert to the banks of the Bega River in a lie-flat seat requires patience and smart routing, since no carrier offers a one-stop miracle here. There is no nonstop business class from Phoenix to Timisoara, and there likely never will be given the market size — instead, the real strategy lies in choosing the right European gateway. Lufthansa routes travelers through Frankfurt or Munich, where its long-haul 747-8 and A350 business class cabins offer generous width and Lufthansa's dependable catering, before a short intra-European hop down to Timisoara. Turkish Airlines, meanwhile, funnels passengers through Istanbul, and I'd argue this is the most interesting option: Turkish's business class lounge in IST is one of the most lavish in the world, complete with a putting green and live cooking stations, making a long layover feel like a destination in itself. Tarom, Romania's flag carrier, doesn't operate widebody service from the US, but it's worth watching for the final domestic-style connection into TSR, since its regional jets run frequently between Bucharest and Timisoara.
Having flown this corridor multiple times, my strongest recommendation is to fly into Munich rather than Frankfurt when connecting on Lufthansa — MUC's passport control and gate proximity for regional European flights is noticeably faster, which matters when your onward Timisoara connection has a tight window. Istanbul routings via Turkish Airlines tend to reward patient travelers with strong award availability in business class, particularly in the shoulder months. Seasonally, January through March and again in October and November are when fares soften considerably, as Romanian leisure demand dries up and business travel to Bucharest and Timisoara's growing automotive and tech sectors becomes the primary driver.
Given how thin and volatile premium inventory can be on this kind of connecting itinerary, I'd strongly suggest setting up automated fare alerts rather than checking manually — the good business class fares on this route tend to appear briefly and vanish fast.