I'm looking into taking a trip to Thailand. However, I won't do that long flight without being in business/first class due to the duration. I have plenty of AA miles to cover the cost but AA doesn't allow you to use them. Any ideas of how to get there using as many AA miles as possible to offset the cost.

Use AA to NRT or HKG or SIN and go from there

You can use them with OneWotld partner airlines I would think Japan airlines fly from Tokyo to Thailand. Or connect through ICN.

What do you mean AA doesn't allow you to use them?

Go to Seats.aero and click the AA Miles dropdown under 'explore'. Then you can filter from North America to Asia and see what is available in business on what days. If you're wanting to travel more than 60 days out, you might need to pay for a subscription for 1 month but in my experience it is well worth it.
It doesn't look like there's any points availability to BKK in the next 60 days but you could get to HND or NRT pretty easily for 60K points.

Depends on your dates but you're absolutely able to use them to travel to that part of the world. You'll almost certainly be connecting in HND or NRT if you're on AA metal, potentially you could fly Cathay as well but they're a bit stingy with partner award space.

My advice when searching for premium-space awards: find your transoceanic leg first, so search from the various AA hubs that fly nonstop to Tokyo (use airport code TYO to search for both Haneda and Narita airports). Once you find the longer leg's award space, it'll be easier to find the connecting leg.

It has been a decade since I've booked that trip.

Your question is a bit vague. I'll interpret what I think your problem is. Since AA doesn't fly into BKK, you can't use aa.com to book the DFW to BKK flight online with points/miles. When you look at flights from DFW to BKK, you are 3 main options, that I've explored.

Overall travel:
DFW - NRT - BKK: Whether it is AA, JAL, or another airline, it is codeshared with AA and you likely can book the DFW-NRT RT with miles.

DFW-DOH-BKK: I think it was Doha/Qatar... Emirates

DFW-<Canada>-BKK: Can't recall if it was Vancouver or Montreal.

Cathay and Emirates are worth the experience. When we flew Cathay... took our kids... our 2yo was a bit of a pill. They calmed him. He fell asleep and when he woke, they were very attentive to ensure he got his meal and snack.

To the points... or miles...
DFW RT To NRT was like 20K NRT to BKK was an additional 10-15K, IIRC.

What we had to do was book the DFW-NRT flights with points. We didn't complete it, IIRC, but put it on "Hold." Then, we had to call AA to get the 2nd part of the leg added. They can add it in their system to your reservation, and then after that we completed the "purchase."

It has been a decade or so, and I may remember wrong or it might have changed, but for us, it was a multi-step process. We did it probably 5-6 times.

If you are having a problem with the 2nd leg, it is likely AA has to actually call the airline to confirm miles, seats, and availability.

You won't want to go through Doha to get to Asia, it would price as two different awards. You can transit North Asia (Asia Region 1) to get to South Asia (Asia Region 2) for one priced award, but not the Middle East to South Asia. (Middle East only transits to India or Africa for "free")

You should be able to book this entire itinerary online, just about every Oneworld partner has their availability on the American site now, it's not like it used to be.