Bradley International isn't a hub anyone associates with Bordeaux, which is precisely why the itinerary rewards a bit of routing savvy. There is no nonstop option, so business class from Hartford to Bordeaux inevitably means a connection, and the most sensible ones run through Paris, Amsterdam-adjacent hubs via partner metal, or Frankfurt. Delta operates BDL to JFK or Boston on regional partners before its transatlantic widebodies take over, and its joint venture with Air France means you can often be ticketed on Delta One out of JFK with a seamless handoff to Air France's connecting flight down to Bordeaux-Mérignac. Air France itself flies the more elegant version of this: connect through Paris-Charles de Gaulle, where its long-haul business cabin—increasingly outfitted with the newer Air France business seat on A350s and 777s—feeds into a short domestic hop on an A220 or A319 down to Bordeaux. Lufthansa's routing through Frankfurt works too, particularly if you're loyal to Star Alliance, though it's the least direct geographically.
My strong preference after flying this corridor repeatedly is the Air France connection through CDG rather than Frankfurt or a second US gateway. Landing in Terminal 2E at CDG puts you close to the Salon Marco Polo lounges, and the domestic-to-Bordeaux leg departs from an adjacent terminal area with minimal walking if you've arrived on a long-haul Air France flight—something not obvious unless you've done the connection in person. Build at least 90 minutes for that transfer regardless of what the ticket suggests, since immigration lines at CDG's Terminal 2E can be unpredictable even for connecting passengers.
Shoulder-season months—January through March, and again in October and November—consistently produce the most favorable business class availability and fares on this city pair, largely because Bordeaux's wine-tourism peak in summer crowds out award and discounted premium space. Given how thin and connection-dependent this route is, setting up automated fare alerts is genuinely the smartest way to catch the rare favorable pricing windows before they close.