Luxembourg remains one of the more delightfully under-the-radar business class destinations for Minneapolis flyers, and the honest answer to how one gets there in comfort is simple: there is no nonstop option, so business class from Minneapolis to Luxembourg means routing through a European hub, most commonly Amsterdam via Delta/KLM or Frankfurt/Munich via Lufthansa, with Luxair providing the final short hop or Lufthansa handling the whole itinerary on its own metal. Delta's Minneapolis strength as a hub works in your favor here—A330 or A350 business class across the Atlantic offers lie-flat suites with direct aisle access, and connecting through Amsterdam-Schiphol is refreshingly efficient compared to the crush of Paris or London. Lufthansa routings through Frankfurt let you experience the airline's business class product, and depending on aircraft assignment you may find yourself in the newer Allegris cabin, though older 747-8 and A340 configurations still fly parts of this network, so it's worth checking your equipment before booking if a specific seat matters to you.
I've routed through both Amsterdam and Frankfurt on this journey, and Frankfurt rewards travelers who arrive with at least 90 minutes of connection time, since Lufthansa's business lounges there (particularly the First Class Lounge, accessible in some fare classes) are worth lingering in even if you're "just" in business. My insider tip: book the Luxair connecting segment as a separate reward or low-cost fare when possible rather than through a single Lufthansa itinerary, since Luxair's short Frankfurt-Luxembourg or Munich-Luxembourg hop rarely has true business class hard product, and paying premium fares for a 45-minute flight is money better spent elsewhere.
Late autumn and the dead of winter—January through March, plus October and November—consistently produce the softest fares on this corridor, as Luxembourg's financial-sector business travel dips outside peak conference seasons. Given how thin scheduling is on this route compared to major transatlantic corridors, setting up fare alerts is genuinely the smartest move you can make before committing to dates.