Transylvania's business hub doesn't yet warrant a nonstop from the Gulf Coast, which means every itinerary in business class from New Orleans to Cluj-Napoca routes through at least one major European gateway. There is no direct service on this pairing; travelers connect via Munich or Frankfurt on Lufthansa, via Istanbul on Turkish Airlines, or via Bucharest on TAROM, with total journey time (including the layover) typically running well beyond the roughly 10.5 hours of actual air time. Of these, I favor the Munich routing on Lufthansa, particularly if you can catch one of the newer Allegris-configured aircraft on the transatlantic leg, with its suite-style doors and generous storage, before transferring to a shorter regional business cabin into Cluj.
Istanbul is the wildcard worth considering seriously. Turkish Airlines' widebody business class, with lie-flat seats and that famously theatrical onboard dining service, makes the long MSY–IST leg genuinely enjoyable, and Istanbul Airport's business lounge — with its indoor putting green and cinema — is worth arriving early for rather than rushing through. From there, the IST–CLJ hop is short, so don't expect a full business product; treat it as a connector flight and save your energy for the long-haul segment. My insider tip: build in a minimum 2.5-hour connection in Istanbul during winter months, since European snow delays regularly cascade through IHA's schedule between December and February, and Turkish's rebooking desks near the business lounge move faster than the general transfer counters.
TAROM's Bucharest routing appeals to purists wanting to stay on one carrier into Romania, though its business cabin is a more modest regional product, so it's best paired with a premium transatlantic partner ticket rather than flown end-to-end. Seasonally, January through March and again in October and November tend to yield the softest business fares on all three routings, as leisure demand to Transylvania dips outside the autumn foliage and Christmas market rushes. Given how these connections and fares shift week to week, setting up automated alerts is the smartest way to catch a genuine business class opening on this route before it closes.