Early 2015 MacBook Air and two external monitors: how to do it and maximum resolution?
Just for posterity I'll answer my own question:

After doing quite a bit of searching around, I eventually figured out that the easiest and cheapest way to accomplish this would be to use the StarTech brand Thunderbolt 2 Dual-Monitor Docking Station. Even with a MacBook Air, you can still plug in two external monitors simultaneously. I found reports that it works as advertised. Note that I have not purchased one and tried it myself, though.