For all effective purposes, you don't

You can upgrade British Airways and Iberia, but only from full fare economy or premium economy. And of course these would be via Atlantic, so prices are probably higher.

I second the idea of premium economy on EVA or Thai (I've flown EVA and thinking about Thai this year since EVA no longer partners with CO), or, if you really do want true Biz, look around some non-US carriers, as they tend to have reasonable Biz fares. A friend of mine who is retired in Thailand used to visit US from BKK on China Air (or Air China - the one based in Taiwan) for very cheap, alas their prices went up. Still should be able to find something sub 5K.

Also, often recommended to try for ethnic travel agents who specialize in travel to Asia. I used a TA in Houston who specializes in Taiwan since EVA is based there, but you can try Korean agents to fly via Seoul as well as other countries.

Hopefully this will help.

EmailKid