Star Alliance member Lufthansa offers fairly priced Business Class flights from Budapest to Seoul available in the next winter season for €1270. These return flights have one (usually) short layover in Frankfurt. If you are flexible and can cheaply reach Budapest this is by far the cheapest option from Europe to South Korea currently available.
Fare Class: Business (P)
Plane(s): B747-400 (FRA-ICN)
Seating: 2-2 or 2-3-2
Travel period: January and February 2024
Travel dates: The fastest is to compare all the cheapest flights on the website of Lufthansa.
Booking sample of Lufthansa Business Class flights to South Korea for €1256! As per the above, your outbound flight from Budapest to Seoul on the 11th of January returning from Asia back to Europe on the 26th of January 2024.