Katowice remains one of Europe's stubbornly nonstop-resistant destinations, and after routing through this corner of Silesia more times than I care to count, I can say with confidence that there is no such thing as a direct path—business class from Phoenix to Katowice always means at least one connection, typically through Amsterdam, Frankfurt, Munich, or Warsaw. The most reliable itinerary I've found pairs Delta's transatlantic metal (a Delta One suite with sliding door on many widebodies) out of Phoenix's Sky Harbor via Atlanta or Detroit to a European gateway, then a final hop on LOT Polish Airlines into Katowice's compact, easy-to-navigate airport. United offers a similar structure via its Polaris cabin, routing through Newark or Chicago before connecting onward, often via Warsaw with LOT completing the journey.
LOT Polish Airlines deserves special mention here, since it's the only carrier of the three with meaningful Polish domestic reach, and its business class on the 787 Dreamliner—lie-flat seats with decent width, though not the most cutting-edge suite in the sky—makes for a comfortable final leg after a long transatlantic push. My insider tip: build in at least a 2.5-hour connection in Warsaw Chopin if using LOT, since immigration and the walk to the domestic gates can eat time that shorter layovers simply don't allow, and missed connections here mean waiting for the next Katowice flight, which isn't frequent. Frankfurt and Munich connections tend to be smoother logistically if your outbound carrier is United or Delta, since both hubs have well-marked, efficient transfer processes for Star Alliance and SkyTeam partners respectively.
Winter and shoulder-season travel—January through March and October into November—consistently deliver the best value on this corridor, partly because Katowice's traffic skews toward Polish diaspora visits and regional business rather than tourism, meaning demand doesn't spike the way it does on Rome or Barcelona routes. Given the multi-carrier, multi-connection nature of this journey, fares fluctuate more than typical single-carrier routes, so setting up price tracking for Phoenix–Katowice business class fares will save you from overpaying on an itinerary that rarely holds steady for long.