business class

Any airline will do it but probably not online. Alitalia is currently flying but their future is in jeopardy. And, it looks like their employees will be going on strike later this month.

Why wouldn't an airline want to sell you these kind of seats?

Are you asking about purchasing a ticket or trying to use miles to book this?

If paying for it then you should have no problems, though this will probably have to be booked as two 1-way tickets. If using miles, I know that United does not allow you to do this for reduced mileage - i.e. they will charge you the full price of the business class mileage even if you fly one way in coach.

I think Delta and Northwest allow you to mix classes on award tickets and BA also does this since it is easy to book one way tickets in their program.

Hello John. You said you do not like an overnight flight to Europe. British air has a day time flight to London LHR airport from New York JFK airport, every day. The airplane departs from New York JFK at 9:10 A.M., and it arrives at London LHR airport in the early evening. Sleep in a hotel near the LHR airport that night, and the next morning fly to your destination in Euorope, via British air. When I inquired about British air flights, they said having a stopover (layover) at London does not cost more money. And British air has a day time flight to London LHR airport from Dulles airport of Washington D.C. (located in Virginia). American airlines has a day time flight to London from Chicago O Hare airport. And, I heard of a day time flight to Paris from New York JFK airport. (I do not know which airline).

When booking here in the UK, the Virgin Atlantic website will let you book one way economy and return by super economy, so maybe you can get business class instead. They fly out NY, I think it's Newark.
I know what you mean about the overnight flight; I dread it too.