Billund's charm—LEGOLAND, minimalist Danish design, a airport small enough to clear in minutes—makes the journey there almost as important as the destination, and there's no nonstop option to complicate matters, since no carrier flies Minneapolis to Billund business class without a stop. My preferred routing has consistently been SAS through Copenhagen, a natural gateway for SAS given Scandinavia is its home turf; the carrier's A330 or A321LR business class on the transatlantic leg offers a private aisle-access design that's held up well against newer competitors, and the short CPH-BLL hop afterward is a breeze even in a tight connection window. Lufthansa via Frankfurt and KLM via Amsterdam are the other realistic paths, each requiring a second connection given neither serves Billund directly with widebody metal—Lufthansa's long-haul business seats vary by aircraft (older 747-8 and A340 configurations persist alongside newer Allegris-equipped 787s and A350s on select routes), so it's worth checking your specific aircraft before booking if seat privacy matters to you.
Having flown all three combinations repeatedly, my strong recommendation is the SAS-only itinerary whenever schedules align, simply because two European connections (as required via FRA or AMS) increase the odds of misconnection and add real hours to a trip that's already a 9.5-hour transatlantic slog before you even reach Jutland. If Frankfurt or Amsterdam pricing is meaningfully better, build in at least 90 minutes for the second connection—both hubs are efficient but sprawling, and KLM's Amsterdam Schiphol transfer between Schengen and non-Schengen piers can eat time unexpectedly during peak summer months.
Late spring through early autumn—May through September—consistently offers the best mix of fare availability and pleasant Danish weather, though this is also peak leisure season for LEGOLAND visitors, so business class award space on SAS tightens noticeably by April. Given how fares fluctuate across three different alliances and routings on this itinerary, setting up automated fare alerts is the smartest way to catch the periodic drops before they disappear.