Louisiana travelers chasing sun on Spain's Canary coast face a genuinely interesting routing puzzle, because there is no single dominant path for business class from New Orleans to Gran Canaria — and that's precisely what makes fare-hunting here rewarding. The honest answer, based on years of watching this corridor, is that you're routing through either Madrid via Iberia, London via British Airways, or Frankfurt/Munich via Lufthansa, since MSY has no widebody nonstops to Europe. Iberia's connection through Madrid is my preferred build: their A330 and A350 business class cabins feature fully lie-flat seats with direct aisle access on the A350, and Madrid-Barajas offers a same-terminal connection at T4S for onward Iberia and Air Europa flights down to LPA, cutting down the layover stress considerably.
British Airways routes New Orleans travelers through London Heathrow, where the Club Suite — BA's much-improved lie-flat product with a closing door — makes the transatlantic leg genuinely comfortable, though the LHR-to-LPA connection typically means a terminal shift and a longer dwell time than the Madrid routing. Lufthansa's option via Frankfurt or Munich brings their long-haul business class (and increasingly the new Allegris seat on select aircraft) into play, appealing to travelers who prefer a German hub with excellent lounge infrastructure, though it's usually the longest overall routing to the Canary Islands geographically. My insider tip: LPA sees a seasonal capacity crunch in winter as Northern Europeans flee for the same sunshine you're after, so shoulder-season travel in April, May, September, or October not only avoids that crowding but consistently yields the strongest business class award and fare availability across all three carriers.
Because pricing on this three-airline, multi-hub route fluctuates with European seasonal demand and Canary Islands tourism patterns, setting up automated fare alerts is far more effective than manually checking — letting the market come to you rather than chasing it.