I would take a look at United for this. Intra-Europe Y fights RT are 30k which enables you to incorporate a stopover and open jaw, which can really help you maximize this kind of travel.

For example you could do the follow:

RDU - JFK - CDG (cash ticket)

Spend time in Paris

CDG - FRA - FCO (stopover)

Spend time in Rome and then travel by train / bus / car around Italy and say end up in Venice

VCE (open jaw) - ZRH - IST (destination - Istanbul is an amazing city and considered part of the Europe region)

Spend time in Istanbul

IST - MUC (layover of less than 24 hours - go to a Biergarten and spend the night) - CDG (return portion of award ticket)

Spend additional time in Paris if you like or take train / car to other parts of France and then return to Paris by that same method

CDG - JFK - RDU (return portion of cash ticket)

Considerations for this approach

1) Finding award space even in Y during the summer can be difficult especially for 4 people. You may have to split up in certain instances and use different routings. For example 2 of you go CDG - FRA - FCO on LH and the other two go CDG - ZRH - FCO on LX. That being said I just did a test for 4 Y tickets CDG - FCO in July and every single day has 4 Y tickets available with some variation of CDG - FRA / MUC / ZRH - FCO routing

2) Stopover and open jaws only apply to roundtrip tickets on UA so this cannot be accomplished using a one-way ticket. This will cost 30k a person so if you end up going with 4 people that's 120k UA. You're most of the way there if you have 50k UA + say a 50/60k Ink signup bonus however you need to sign up soon since the minimum spend will need to met relatively quickly so you can have the necessary UR points post and transferred to UA so you can book for this summer