Researching flights to Viet Nam

My first priorities are always price and flight time/connections/convenience. Flying from Chicago to Istanbul and Istanbul to Vietnam will add hours of travel time, so it would not be my first choice.

I've never flown EVA Air, but I see they have a flight via Taipei (Taiwan) with only that one connection, and reasonable fares, so that's probably how I'd go.

Check it out on Google Flights.

I totally agree with the criteria you use in researching flights. My only thought was with a 9 1/2 hour layover, it might be possible to leave the airport for a peak at the city. Thoughts?

Sure, you could try to take advantage of the layover and pop into Istanbul for a few hours, if the schedule works out. I've never done that. Just get a visa ahead of time:

https://tr.usembassy.gov/u-s-citizen-services/u-s-citizen-services-faqs/

Personal preference would be just to get to my destination ASAP rather than prolong it with a detour - and visit Turkey some other time...or try to extend the layover for a few days, to make the stop worthwhile rather than a few blurred hours between time zones.

I've been to Istanbul in 2015, so not really recently. It was my 3rd time there (2 solo trips and one RS group tour). I would look into the "Touristanbul" layover program through Turkish Airlines if the timing works out for you (depending on when exactly you'd get into Istanbul). It's a very innovative introduction to the country and risk-free. I highly recommend Turkish Airlines - great food, service, and prices. The Turkish people are very hospitable.
https://www.turkishairlines.com/en-gr/flights/fly-different/touristanbul/what-is-touristanbul/index.html

There's just no easy way to get to SE Asia, even from the left coast. Personally, I would not fly through Istanbul, especially since the visa situation has been fluid.. There is a flight to Tokyo Narita from Chicago. l forgot when you're planning to go, but I did some dummy dates on Google Flights ORD-SGN and the prices are actually good (in December anyway, $885 round trip in coach).

I flew Korean Airlines from Portland - Seattle - Seoul - Bankgok and Hanoi-Vancouver-Portland on way back. I was in cattle class but really enjoyed the service (great flight staff) and the food was, amazingly, quite good (especially the Korean fare). One thing you might consider is flying to a West Coast hub (LA, San Fran, Seattle, or Vancouver) and spending the night on the way. You'd get a couple hours on the jet lag/time change and possibly a better fare.

Turkish Airlines is great. If you are transiting in Istanbul (and you meet their criteria - see the link above), they offer you your choice of either a night in a nice hotel, or a free tour of the city. What other airline would ever dream of offering such perks?

That said, you would be adding to the length of your trip. If you had a strong interest in getting a quickie look at Istanbul, that might tip the scales, but if your goal is Vietnam, I'd head for Vietnam. Keep in mind that, unless the time for your trip is unconstrained, any time spent en route and/or at an intermediate stop comes out of your time spent at your primary destination - do you have enough days there or would you be skipping or squeezing things to add Istanbul? That said, when I can pull it off, I always like to add on a short stay in someplace else I might not otherwise have the chance to see.

Your research will go faster if you use an on-line agency to sort through the many possible combinations, either east or west, rather than trying to piece them together yourself. No site covers all airlines but I think www.matrix.itasoftware is bigger than most (it doesn't sell tickets but the info is useful.) Vietnam Airlines serves a number of European gateways so you could get a combined ticket if you want to go east. More possibilities will come up flying west, often with a stop in Japan but the Korean airlines will take you through Seoul or Cathay Pacific goes through Hong Kong. Looking at next spring, some of the fares are remarkably cheap considering the long distances. I also think you should explore flying into Ho Chi Minh and home from Hanoi (or the reverse) using a multi-destination search function.
It's worth doing the work as Vietnam is a fascinating destination and the food is great. Cheap, too.

I flew Turkish from Tel Aviv to Chicago this summer and spent 5 nights in Istanbul (second visit) on the outbound. This month I flew Turkish from Tel Aviv to Casablanca. I find Turkish very comfortable in coach with a little more room than other airlines, good service and good food. Each seat has a personal screen with a large selection of movies, tv programs, and live tv (mostly news stations, BBC, CNN, etc). There's no problem going through the airport in Istanbul. I believe you can pay for entrance to the business lounge even if you don't fly business.

Where are you flying to/from in SE Asia? I flew JetStar, Vietjet and Thai Air. Flights are short so even though they may not be comfortable, it's over soon :-) Thai Air was very nice.