Memphis doesn't hand you a nonstop path to Legoland's hometown, but the connections are gentler than you'd expect, and business class from Memphis to Billund can actually be one of the more painless transatlantic itineraries out of the Mid-South if you route it correctly. There's no direct service on this pairing, so your real decision is choosing a hub: Amsterdam via KLM, Frankfurt or Munich via Lufthansa, or Copenhagen via SAS, with the last option being my preferred routing since Billund sits close enough to Copenhagen that SAS effectively owns the local knowledge on this corner of Jutland. SAS flies A321LR and A330 equipment on many of its US gateways with a proper lie-flat business product, and their short intra-European hop down to Billund keeps connection times sane at Kastrup.
Having flown all three carriers into Scandinavia repeatedly, I'd steer discerning travelers toward SAS's Copenhagen routing or Lufthansa's Frankfurt connection if schedule timing favors it — Lufthansa's long-haul business seats, especially on 747-8 and A350 equipment depending on gateway, remain a dependable, if not flashy, product, while KLM's World Business Class on its 787 and A330 fleet offers a warmer, more personal service style that suits Scandinavian-bound leisure travelers well. My insider tip: if connecting through Amsterdam, request an early morning MEM departure whenever schedules allow, since it lines up with KLM's Billund frequency and avoids the dreaded overnight layover that many itineraries otherwise force onto this route.
Late spring through early autumn — May through September — is when Denmark rewards the effort, with long daylight hours and reliable weather for exploring Jutland beyond the Legoland gates. Award and premium cash fares on this corridor fluctuate more than travelers expect given the modest local demand, so I'd recommend setting up fare alerts and tracking pricing trends over several weeks before booking, rather than committing to the first business class fare you see.